Question 12 Marks
Lenz's law is a specific statement of law of conservation of energy-Explain.
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Question 22 Marks
What is self induction ? Obtain expression for self induced emf.
Answer
View full question & answer→→When electric current through an isolated conducting coil is changed, magnetic flux linked with it changes. As a result, induced emf is produced in the coil. This phenomenon is called self induction.
→Here, induced emf is called self induced emf.
→Suppose, electric current passing through an isolated coil having N turns is I .
→Total magnetic flux linked with coil,
$\begin{array}{c}
N \phi_{ B } \propto I \\
\therefore \quad N \phi_{ B }= L I
\end{array}$
→Proportionality constant L in equation (1) is called self inductance.
→On changing current with time, magnetic flux linked changes. As a result, induced emf is produced.
$\therefore N \frac{d \phi_{ B }}{d t}= L \frac{d I }{d t}$
→According to Faraday's law,
$\varepsilon=- N \frac{d \phi_{ B }}{d t}$
From equation (2) and (3),
$\varepsilon=- L \frac{d I }{d t}$
Equation (4) is expression for self induced emf.
→Here, induced emf is called self induced emf.
→Suppose, electric current passing through an isolated coil having N turns is I .
→Total magnetic flux linked with coil,
$\begin{array}{c}
N \phi_{ B } \propto I \\
\therefore \quad N \phi_{ B }= L I
\end{array}$
→Proportionality constant L in equation (1) is called self inductance.
→On changing current with time, magnetic flux linked changes. As a result, induced emf is produced.
$\therefore N \frac{d \phi_{ B }}{d t}= L \frac{d I }{d t}$
→According to Faraday's law,
$\varepsilon=- N \frac{d \phi_{ B }}{d t}$
From equation (2) and (3),
$\varepsilon=- L \frac{d I }{d t}$
Equation (4) is expression for self induced emf.
Question 32 Marks
Write two definitions of mutual inductance. State what it depends on and state its unit and dimensional formula.
Answer
View full question & answer→$\begin{array}{c}\text { (I) } \varepsilon_2=- M _{21} \frac{d I _1}{d t} \\
\text { If } \frac{d I _1}{d t}=1 \frac{ A }{ S } \\
\text { so, } \varepsilon_2=- M _{21}\end{array}$
→"In a system consisting of two windings; if the rate of change of current in one winding is unity, the emf induced in the other winding is called mutual inductance of the two winding system."
(II) $\phi_2= M _{21} I _1$
If $I _1=1 A$ thus $\phi_2= M _{21}$
→"In a system consisting of two windings; when the current through one winding is unity the magnetic flux associated with the other winding is called the mutual inductance of the system of two winding.
• Mutual inductance depends on the following factors :
(1) shape of coil
(2) their dimensions
(3) Number of turns
(4) Magnetic properties of the medium between the coil
(5) Their relative bowing
(6) Their relative distance.
→Unit : $H , \frac{ Wb }{ A }, \frac{ V s}{A}$
→Dimensional Formula : $M ^1 L^2 T^{-2} A^{-2}$
\text { If } \frac{d I _1}{d t}=1 \frac{ A }{ S } \\
\text { so, } \varepsilon_2=- M _{21}\end{array}$
→"In a system consisting of two windings; if the rate of change of current in one winding is unity, the emf induced in the other winding is called mutual inductance of the two winding system."
(II) $\phi_2= M _{21} I _1$
If $I _1=1 A$ thus $\phi_2= M _{21}$
→"In a system consisting of two windings; when the current through one winding is unity the magnetic flux associated with the other winding is called the mutual inductance of the system of two winding.
• Mutual inductance depends on the following factors :
(1) shape of coil
(2) their dimensions
(3) Number of turns
(4) Magnetic properties of the medium between the coil
(5) Their relative bowing
(6) Their relative distance.
→Unit : $H , \frac{ Wb }{ A }, \frac{ V s}{A}$
→Dimensional Formula : $M ^1 L^2 T^{-2} A^{-2}$
Question 42 Marks
Obtain expression for energy required to establish electric current I in a coil having self inductance L.
###
Obtain expression $U_B=\frac{1}{2} L I ^2$ for magnetic energy stored while establishing current I in coil having self inductance L.
###
Obtain expression $U_B=\frac{1}{2} L I ^2$ for magnetic energy stored while establishing current I in coil having self inductance L.
Answer
View full question & answer→→Self induced emf in a coil having self inductance L is
$\varepsilon=- L \frac{d I }{d t}$
This self induced emf opposes the change in current taking place in coil. Hence it is also called Back emf.
→Physically, self inductance plays the role of inertia in electricity.
→Work is required to be done against back emf to establish electric current in coil. This energy spent gets stored in form of magnetic energy $U _{ B }$ in the coil.
→Suppose, time rate of work done to establish current I in coil at any instant is $\frac{d W}{d t}$ then
$\begin{aligned}
\frac{d W}{d t} & =|\varepsilon| I \text { (neglecting ohmic loss.) } \\
\therefore \quad \frac{d W}{d t} & = L I \frac{d I }{d t} \text { (from equation (1)) }
\end{aligned}$
→Total work done to establish current I
$\begin{aligned}
W & =\int_0^1 d W=\int_0^1 LI d I = L \int_0^1 I d I \\
\therefore \quad W & =\frac{1}{2} L I ^2
\end{aligned}$
→Energy spent in doing this work gets stored in form of magnetic energy in the coil.
$\therefore$ Magnetic potential energy $U _{ B }=\frac{1}{2} LI ^2 \ldots$
$\varepsilon=- L \frac{d I }{d t}$
This self induced emf opposes the change in current taking place in coil. Hence it is also called Back emf.
→Physically, self inductance plays the role of inertia in electricity.
→Work is required to be done against back emf to establish electric current in coil. This energy spent gets stored in form of magnetic energy $U _{ B }$ in the coil.
→Suppose, time rate of work done to establish current I in coil at any instant is $\frac{d W}{d t}$ then
$\begin{aligned}
\frac{d W}{d t} & =|\varepsilon| I \text { (neglecting ohmic loss.) } \\
\therefore \quad \frac{d W}{d t} & = L I \frac{d I }{d t} \text { (from equation (1)) }
\end{aligned}$
→Total work done to establish current I
$\begin{aligned}
W & =\int_0^1 d W=\int_0^1 LI d I = L \int_0^1 I d I \\
\therefore \quad W & =\frac{1}{2} L I ^2
\end{aligned}$
→Energy spent in doing this work gets stored in form of magnetic energy in the coil.
$\therefore$ Magnetic potential energy $U _{ B }=\frac{1}{2} LI ^2 \ldots$
Question 52 Marks
What is Electromagnetic induction ? Give names of experiments of Faraday and Henry explaining this phenomenon.
Answer
View full question & answer→→Electric current is formed in closed circuits by changing magnetic fields. This phenomenon is called Electromagnetic induction.
→Following are the experiments done by Faraday and Henry explaining phenomenon of Electromagnetic induction :
(i) Experiment of relative motion between magnet and coil.
(ii) Experiment of relative motion between two coils.
(iii) Experiment of two stationary coils.
→Following are the experiments done by Faraday and Henry explaining phenomenon of Electromagnetic induction :
(i) Experiment of relative motion between magnet and coil.
(ii) Experiment of relative motion between two coils.
(iii) Experiment of two stationary coils.
Question 62 Marks
State types of commercial generators.
Answer
View full question & answer→→Electrical energy is generated commercially in commercial generators. Different types of generators are as follows.
(i) Hydro-electric generator :
→The mechanical energy required for rotation of armature in this generator is obtained from water falling from a height. ex dams.
(ii) Thermal generators :
→Water is heated to produce steam using coal or other sources. The steam at high pressure produces rotation of armature.
(iii) Nuclear power generators :
→Instead of coal, if nuclear fuel is used then such generators are called Nuclear power generators.
(i) Hydro-electric generator :
→The mechanical energy required for rotation of armature in this generator is obtained from water falling from a height. ex dams.
(ii) Thermal generators :
→Water is heated to produce steam using coal or other sources. The steam at high pressure produces rotation of armature.
(iii) Nuclear power generators :
→Instead of coal, if nuclear fuel is used then such generators are called Nuclear power generators.
Question 72 Marks
Obtain expression for induced emf in any one coil due to electric currents flowing at same time in two coils placed near each other.
Answer
View full question & answer→→When electric currents are passed simultaneously through two coils placed near each other, magnetic flux linked with any one coil is sum of individual magnetic flux linked independently with both coils.
→Suppose, electric currents passing simultaneously through two nearby coils having turns $N _1$ and $N _2$ is $I _1$ and $I _2$ respectively.
→Total magnetic flux linked with coil having $N _1$ turns is
$\therefore N _1 \phi_1= M _{11} I _1+ M _{12} I _2$
Where $M _{11}$ inductance due to first coil
$\begin{array}{ll}
\therefore \quad & M _{11}= L _1 \quad \text { and } \\
& M _{12}=\text { mutual inductance of first coil } \\
& \text { with respect to second coil } \\
\therefore \quad N _1 \phi_1= & L _1 I _1+ M _{12} I _2
\end{array}$
According to Faraday's law, induced emf in first coil,
$\begin{aligned}
\varepsilon_1 & =- N _1 \frac{d \phi_1}{d t} \\
\therefore \varepsilon_1 & =- L _1 \frac{d I _1}{d t}- M _{12} \frac{d I _2}{d t}
\end{aligned}$
Equation (3) is required expression.
→Suppose, electric currents passing simultaneously through two nearby coils having turns $N _1$ and $N _2$ is $I _1$ and $I _2$ respectively.
→Total magnetic flux linked with coil having $N _1$ turns is
$\therefore N _1 \phi_1= M _{11} I _1+ M _{12} I _2$
Where $M _{11}$ inductance due to first coil
$\begin{array}{ll}
\therefore \quad & M _{11}= L _1 \quad \text { and } \\
& M _{12}=\text { mutual inductance of first coil } \\
& \text { with respect to second coil } \\
\therefore \quad N _1 \phi_1= & L _1 I _1+ M _{12} I _2
\end{array}$
According to Faraday's law, induced emf in first coil,
$\begin{aligned}
\varepsilon_1 & =- N _1 \frac{d \phi_1}{d t} \\
\therefore \varepsilon_1 & =- L _1 \frac{d I _1}{d t}- M _{12} \frac{d I _2}{d t}
\end{aligned}$
Equation (3) is required expression.
Question 82 Marks
Obtain expression of self inductance of a solenoid having length $l$ and cross-section of area A .
Answer
View full question & answer→→For a solenoid having length $l$ and area of cross-section A,
$I =$ electric current passed
$N =$ total number of turns
$n=$ number of turns per unit length
$\therefore N =n l$
→Magnetic field produced in solenoid,
$B =\mu_0 n I$
→Total magnetic flux linked,
$N \phi_{ B }=(n l) AB =(n l)( A )\left(\mu_0 n I \right)$
→Self inductance of solenoid,
$L =\frac{ N \phi_{ B }}{ I }$
From equation (1) & (2),
$\begin{aligned}
L & =\frac{\mu_0 n^2 A l I }{ I } \\
\therefore \quad L & =\mu_0 n^2 A l
\end{aligned}$
→Equation (3) is expression for self inductance of solenoid.
→If substance having relative permeability $\mu_r$ is filled inside the solenoid then, self inductance $L =\mu_r \mu_0 n^2 A l$
→Self inductance of coil depends on its shape, size and permeability of medium on which coil is wound.
$I =$ electric current passed
$N =$ total number of turns
$n=$ number of turns per unit length
$\therefore N =n l$
→Magnetic field produced in solenoid,
$B =\mu_0 n I$
→Total magnetic flux linked,
$N \phi_{ B }=(n l) AB =(n l)( A )\left(\mu_0 n I \right)$
→Self inductance of solenoid,
$L =\frac{ N \phi_{ B }}{ I }$
From equation (1) & (2),
$\begin{aligned}
L & =\frac{\mu_0 n^2 A l I }{ I } \\
\therefore \quad L & =\mu_0 n^2 A l
\end{aligned}$
→Equation (3) is expression for self inductance of solenoid.
→If substance having relative permeability $\mu_r$ is filled inside the solenoid then, self inductance $L =\mu_r \mu_0 n^2 A l$
→Self inductance of coil depends on its shape, size and permeability of medium on which coil is wound.
Question 92 Marks
Write the definition of self inductance. What is depends on and state its unit and dimensional formula.
Answer
View full question & answer→→Induced emf $\varepsilon=- L \frac{d I }{d t}$
If $\frac{d I }{d t}=1 \frac{ A }{ s }$
so, $\varepsilon=- L$
→"If the rate of change of current passing through the coil is unit, then the emf induced in the coil is called self-induced emf and this phonomenon is called self inductance of the coil.
• Self inductance depends on the following factors :
(1) Dimensions of coil
(2) Shape and number of turns in coil
(3) Magnetic properties of medium.
→Unit : henry $( H ), \frac{ W b}{A}, \frac{ V s}{A}$
→Dimensional Formula : $M ^1 L^2 T^{-2} A^{-2}$
If $\frac{d I }{d t}=1 \frac{ A }{ s }$
so, $\varepsilon=- L$
→"If the rate of change of current passing through the coil is unit, then the emf induced in the coil is called self-induced emf and this phonomenon is called self inductance of the coil.
• Self inductance depends on the following factors :
(1) Dimensions of coil
(2) Shape and number of turns in coil
(3) Magnetic properties of medium.
→Unit : henry $( H ), \frac{ W b}{A}, \frac{ V s}{A}$
→Dimensional Formula : $M ^1 L^2 T^{-2} A^{-2}$
Question 102 Marks
Discuss induced emf produced in a conducting rod during its motion in direction perpendicular to uniform magnetic field with reference to Lorentz force and obtain its expression.
Answer
View full question & answer→Force acting on free electron is $\overrightarrow{ F }=-e(\vec{v} \times \overrightarrow{ B })$. Due to this force, free electron tend to meet at end Q and same amount of positive charge gets exposed at end $P$. Potential difference arising between P and Q has value $B vl$. Thus, it acts like a battery having emf $\varepsilon= Bvl$.
Question 112 Marks
State history of "Discovery of Electromagnetic Induction'' and its contribution in progress of today's civilisation.
Answer
View full question & answer→→From experiments on electric current done by Oersted, Ampere and other scientists, it was established that electricity and magnetism are inter-related.
→"Magnetic needle placed near current carrying wire shows deflection." Reverse of this process, 'can moving magnets produce electric current ?'
→Affirmative reply to this question was first obtained from experiments done by Michael
Faraday and Henry Joseph and electromagnetic induction was discovered.
→From their experiments, it was proved that "changing magnetic fields can produce electric current." This phenomenon is called Electromagnetic Induction.
→Direct Contribution of Electromagnetic induction is seen in the progress of modern generators and transformers. This shows importance of electromagnetic induction in progress of present civilisation.
→"Magnetic needle placed near current carrying wire shows deflection." Reverse of this process, 'can moving magnets produce electric current ?'
→Affirmative reply to this question was first obtained from experiments done by Michael
Faraday and Henry Joseph and electromagnetic induction was discovered.
→From their experiments, it was proved that "changing magnetic fields can produce electric current." This phenomenon is called Electromagnetic Induction.
→Direct Contribution of Electromagnetic induction is seen in the progress of modern generators and transformers. This shows importance of electromagnetic induction in progress of present civilisation.
Question 122 Marks
Describe experiment of two stationary coil which explains clectromagnetic induction.
Answer
→Even if there is no relative motion between two coils, phenomenon of electromagnetic induction can be obtained.
→Two stationary coils $C _1$ and $C _2$ are shown in figure.
→Galvanometer is connected to $C _1$ and battery and key K are connected to $C _2$.
→On pressing K (closing), galvanometer shows momentary deflection & then comes to zero.
→After pressing (closing) key and keeping it in same situation, galvanometer does not show deflection.
→On releasing key (opening), galvanometer shows deflection in opposite direction.
→On repeating experiment keeping iron rod on axis of coil, very large deflection is seen in galvanometer.
→Thus, it is proved that a changing magnetic field produces electric current.
View full question & answer→→Even if there is no relative motion between two coils, phenomenon of electromagnetic induction can be obtained.
→Two stationary coils $C _1$ and $C _2$ are shown in figure.
→Galvanometer is connected to $C _1$ and battery and key K are connected to $C _2$.
→On pressing K (closing), galvanometer shows momentary deflection & then comes to zero.
→After pressing (closing) key and keeping it in same situation, galvanometer does not show deflection.
→On releasing key (opening), galvanometer shows deflection in opposite direction.
→On repeating experiment keeping iron rod on axis of coil, very large deflection is seen in galvanometer.
→Thus, it is proved that a changing magnetic field produces electric current.
Question 132 Marks
Describe experiment of two coils explaining electromagnetic induction.
Answer
→As shown in figure, two coils $C _1$ and $C _2$ are kept facing each other.
→Galvanometer is connected with $C _1$ and Battery is connected with $C _2$.
→On passing steady current through coil $C _2$, magnetic field is produced around it.
→On moving $C _2$ towards $C _1$, galvanometer shows deflection. It indicates that electric current is induced in $C _1$.
→On moving $C _2$ away from $C _1$, galvanometer shows deflection in opposite direction.
→Keeping $C _2$ stationary, on moving $C _1$ closer to or away from $C _2$, we get same results as above.
→Thus, here electric current is induced due to relative motion between both coils.
View full question & answer→→As shown in figure, two coils $C _1$ and $C _2$ are kept facing each other.
→Galvanometer is connected with $C _1$ and Battery is connected with $C _2$.
→On passing steady current through coil $C _2$, magnetic field is produced around it.
→On moving $C _2$ towards $C _1$, galvanometer shows deflection. It indicates that electric current is induced in $C _1$.
→On moving $C _2$ away from $C _1$, galvanometer shows deflection in opposite direction.
→Keeping $C _2$ stationary, on moving $C _1$ closer to or away from $C _2$, we get same results as above.
→Thus, here electric current is induced due to relative motion between both coils.
Question 142 Marks
Describe experiment of relative motion between magnet and coil explaining phenomenon of electromagnetic induction.
Answer
→As shown in figure, coil $C _1$ made of conducting material and having a coating of non-conducting material is connected to a galvanometer ( G ).
→A bar magnet is placed such that N -pole of bar magnet faces coil $C_1$.
→On moving N -pole of bar magnet towards coil, galvanometer shows deflection. This deflection is seen till magnet is in motion.
→When magnet is stationary, no deflection is seen in galvanometer.
→Instead of N-pole, if S-pole of bar magnet is kept facing the coil and moved towards coil, galvanometer shows deflection but in opposite direction than before.
→Keeping magnet stationary, if coil is moved towards or away from magnet, then in both cases, galvanometer shows deflection in opposite directions.
→On increasing speed of relative motion between both, galvanometer shows more deflection and hence electric currents are produced.
→Clear conclusion from above experiment is that when there is a relative motion between coil and magnet, then only galvanometer shows deflection and electric current is produced in coil.
→Thus, relative motion between coil and magnet is responsible for production of electric current.
View full question & answer→→As shown in figure, coil $C _1$ made of conducting material and having a coating of non-conducting material is connected to a galvanometer ( G ).
→A bar magnet is placed such that N -pole of bar magnet faces coil $C_1$.
→On moving N -pole of bar magnet towards coil, galvanometer shows deflection. This deflection is seen till magnet is in motion.
→When magnet is stationary, no deflection is seen in galvanometer.
→Instead of N-pole, if S-pole of bar magnet is kept facing the coil and moved towards coil, galvanometer shows deflection but in opposite direction than before.
→Keeping magnet stationary, if coil is moved towards or away from magnet, then in both cases, galvanometer shows deflection in opposite directions.
→On increasing speed of relative motion between both, galvanometer shows more deflection and hence electric currents are produced.
→Clear conclusion from above experiment is that when there is a relative motion between coil and magnet, then only galvanometer shows deflection and electric current is produced in coil.
→Thus, relative motion between coil and magnet is responsible for production of electric current.
Question 152 Marks
Explain Magnetic flux with figure.
Answer
• Magnetic Flux :
→Number of magnetic field lines passing perpendicularly through a surface kept in magnetic field is called Magnetic flux $\left(\phi_{ B }\right.$ ) linked with that surface.
→In figure (a), Magnetic flux passing through plane having area A placed in a uniform magnetic field $\vec{B}$ can be given as follows.
$\phi_{ B }=\overrightarrow{ B } \cdot \overrightarrow{ A }= BA \cos \theta$
Where, $\overrightarrow{ A }=$ Area Vector
which is shown perpendicular to plane.
$\theta$ is angle between $\overrightarrow{ A }$ and $\overrightarrow{ B }$
→Unit : W $b$ (weber) OR Tm²
→Dimensions : $M ^1 L^2 T^{-2} A^{-1}$
→As shown in figure (b), when $\vec{B}$ and $\vec{A}$ have different magnitudes and directions at different
parts of plane (when there is non-uniform magnetic field and curved surface), then magnetic flux passing through surface can be obtained in following manner :
$\begin{aligned}
\therefore \phi_{ B } & =\overrightarrow{ B }_1 \cdot d \overrightarrow{ A _1}+\overrightarrow{ B _2} \cdot d \overrightarrow{ A _2}+\ldots . . \\
& =\sum_{\text {all }} \overrightarrow{ B _i} \cdot d \overrightarrow{ A _i}
\end{aligned}$
It means summation of magnetic flux passing through all surface elements. Where $\overrightarrow{d A_i}$ is $i^{\text {th }}$ area element and $\overrightarrow{ B }_i$ magnetic field at area element $d \overrightarrow{ A _i}$.
View full question & answer→• Magnetic Flux :
→Number of magnetic field lines passing perpendicularly through a surface kept in magnetic field is called Magnetic flux $\left(\phi_{ B }\right.$ ) linked with that surface.
→In figure (a), Magnetic flux passing through plane having area A placed in a uniform magnetic field $\vec{B}$ can be given as follows.
$\phi_{ B }=\overrightarrow{ B } \cdot \overrightarrow{ A }= BA \cos \theta$
Where, $\overrightarrow{ A }=$ Area Vector
which is shown perpendicular to plane.
$\theta$ is angle between $\overrightarrow{ A }$ and $\overrightarrow{ B }$
→Unit : W $b$ (weber) OR Tm²
→Dimensions : $M ^1 L^2 T^{-2} A^{-1}$
→As shown in figure (b), when $\vec{B}$ and $\vec{A}$ have different magnitudes and directions at different
parts of plane (when there is non-uniform magnetic field and curved surface), then magnetic flux passing through surface can be obtained in following manner :
$\begin{aligned}
\therefore \phi_{ B } & =\overrightarrow{ B }_1 \cdot d \overrightarrow{ A _1}+\overrightarrow{ B _2} \cdot d \overrightarrow{ A _2}+\ldots . . \\
& =\sum_{\text {all }} \overrightarrow{ B _i} \cdot d \overrightarrow{ A _i}
\end{aligned}$
It means summation of magnetic flux passing through all surface elements. Where $\overrightarrow{d A_i}$ is $i^{\text {th }}$ area element and $\overrightarrow{ B }_i$ magnetic field at area element $d \overrightarrow{ A _i}$.
Question 162 Marks
State and explain Lenz's law
Question 172 Marks
Obtain Faraday's law of induction explaining Faraday's experiments regarding electric induction with reference to magnetic flux.
Answer
View full question & answer→→From experimental observations of experiments of Faraday and Henry, it is clear that initially magnetic flux gets linked with coil, after that linked magnetic flux changes and then induced current or induced emf arises.
•Experiment 1 :
→In experiment of coil and magnet, magnetic flux is linked with coil due to magnet.
→On giving relative motion between them, linked magnetic flux changes and induced current arises.
• Experiment 2 :
→In experiment of two coils, on passing electric current through any one coil, magnetic field is produced. Due to this, magnetic flux is linked with other coil.
→On giving relative motion between two coils, linked magnetic flux changes and induced current is produced.
• Experiment 3 :
→In experiment of two stationary coils, making key on-off, produces magnetic field and hence magnetic flux linked with other coil changes and thus galvanometer shows deflection.
→ Keeping key in closed situation, linked magnetic flux does not change and galvanometer shows zero deflection.
→Thus, clear conclusion from all these experiments is that linking magnetic flux with coil by any means, changing linked magnetic flux with time, induced emf is produced.
• Faraday's law of electromagnetic induction :
→The magnitude of induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit.
→Induced emf is shown mathematically as below :
$\varepsilon=-\frac{d \phi_{ B }}{d t}$
→The negative sign in the equation indicates the direction of $\varepsilon$ and hence the direction of current in the closed circuit (loop). (It can also be said that it shows presence of Lenz's law.)
→ If number of turns in coil is N then
$\varepsilon=- N \frac{d \phi_{ B }}{d t}$
•Experiment 1 :
→In experiment of coil and magnet, magnetic flux is linked with coil due to magnet.
→On giving relative motion between them, linked magnetic flux changes and induced current arises.
• Experiment 2 :
→In experiment of two coils, on passing electric current through any one coil, magnetic field is produced. Due to this, magnetic flux is linked with other coil.
→On giving relative motion between two coils, linked magnetic flux changes and induced current is produced.
• Experiment 3 :
→In experiment of two stationary coils, making key on-off, produces magnetic field and hence magnetic flux linked with other coil changes and thus galvanometer shows deflection.
→ Keeping key in closed situation, linked magnetic flux does not change and galvanometer shows zero deflection.
→Thus, clear conclusion from all these experiments is that linking magnetic flux with coil by any means, changing linked magnetic flux with time, induced emf is produced.
• Faraday's law of electromagnetic induction :
→The magnitude of induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit.
→Induced emf is shown mathematically as below :
$\varepsilon=-\frac{d \phi_{ B }}{d t}$
→The negative sign in the equation indicates the direction of $\varepsilon$ and hence the direction of current in the closed circuit (loop). (It can also be said that it shows presence of Lenz's law.)
→ If number of turns in coil is N then
$\varepsilon=- N \frac{d \phi_{ B }}{d t}$
Question 182 Marks
Explain mutual induction using Faraday's two coil experiment. Obtain expression
Answer
View full question & answer→$\varepsilon_1=- M \frac{d l_2}{d t}$.
→In the figure, arrangement of two coil experiment is shown. Here, coils $C _1$ and $C _2$ are placed coaxially.
→When electric current $I _2$ passing through coil $C _2$ changes, magnetic flux linked with coil $C _1$ changes and hence induced emf $\varepsilon_1$ arises.
→When electric current $I _2$ flows through $C _2$, total magnetic flux linked with coil $C _1$ having $N _1$ turns is $N _1 \phi_1 \propto I _2$
$\therefore N _1 \phi_1= M I _2$
Where, M is mutual inductance
→For current changing with time,
$\frac{d}{d t}\left(N_1 \phi_1\right)= M \frac{d}{d t}\left( I _2\right)$
→According to Faraday's law, induced emf in $C _1$
$\varepsilon_1=-\frac{d}{d t}\left(N_1 \phi_1\right)$
→From equation (2) and (3),
$\varepsilon_1=- M \frac{d I _2}{d t}$
→Equation (4) is the required result.
→Here, value of induced emf depends on time rate of change of electric current and mutual inductance.
→In the figure, arrangement of two coil experiment is shown. Here, coils $C _1$ and $C _2$ are placed coaxially.
→When electric current $I _2$ passing through coil $C _2$ changes, magnetic flux linked with coil $C _1$ changes and hence induced emf $\varepsilon_1$ arises.
→When electric current $I _2$ flows through $C _2$, total magnetic flux linked with coil $C _1$ having $N _1$ turns is $N _1 \phi_1 \propto I _2$
$\therefore N _1 \phi_1= M I _2$
Where, M is mutual inductance
→For current changing with time,
$\frac{d}{d t}\left(N_1 \phi_1\right)= M \frac{d}{d t}\left( I _2\right)$
→According to Faraday's law, induced emf in $C _1$
$\varepsilon_1=-\frac{d}{d t}\left(N_1 \phi_1\right)$
→From equation (2) and (3),
$\varepsilon_1=- M \frac{d I _2}{d t}$
→Equation (4) is the required result.
→Here, value of induced emf depends on time rate of change of electric current and mutual inductance.
Question 192 Marks
Give expression for induced emf in AC generator. Discuss, with fogure, how this emf changes with time during one rotation of coil.
Answer
View full question & answer→→Expression for induced emf in AC generator is
$\varepsilon=\varepsilon_0 \sin \omega t$
where $\varepsilon_0= NBA \omega$
→Induced emf changes periodically with time during one complete rotation. This is shown step-wise in following figure of graph of emf $\varepsilon \rightarrow$ time $t$. Time $t$ is in the form of periodic time T.
→It is clear that when $\theta=\frac{\pi}{2}$ or $\frac{3 \pi}{2}$, induced emf is maximum.
→When $\theta=0$ or $\pi, 2 \pi$ induced emf is zero.
→This situation is repeated till rotation of coil continues and induced emf $\varepsilon$ is produced between $+\varepsilon_0$ and $-\varepsilon_0$.
$\varepsilon=\varepsilon_0 \sin \omega t$
where $\varepsilon_0= NBA \omega$
→Induced emf changes periodically with time during one complete rotation. This is shown step-wise in following figure of graph of emf $\varepsilon \rightarrow$ time $t$. Time $t$ is in the form of periodic time T.
→It is clear that when $\theta=\frac{\pi}{2}$ or $\frac{3 \pi}{2}$, induced emf is maximum.
→When $\theta=0$ or $\pi, 2 \pi$ induced emf is zero.
→This situation is repeated till rotation of coil continues and induced emf $\varepsilon$ is produced between $+\varepsilon_0$ and $-\varepsilon_0$.
Question 202 Marks
Explain Inductance. Give its unit and dimensions.
Answer
View full question & answer→→Induced current is formed when magnetic flux linked with any coil is changed,
(i) by changing electric current passing through coil itself
OR
(ii) by changing electric current passing through another coil placed nearby.
→In above both cases, magnetic flux $\phi_{ B }$ linked with coil is directly proportional to electric current I .
$\therefore \phi_{ B } \propto I$
→If dimensions of coil do not change with time, then
$\frac{d \phi_{ B }}{d t} \propto \frac{d I }{d t}$
→If coil has N closely wound turns, then total magnetic flux linked will be $N \phi_{ B }$ which is
also indirectly proportional to electric current passing through coil.
$N \phi_{ B } \alpha I$
The constant of proportionality in equation (1) & (3) is called Inductance.
→Inductance is a scalar quantity. It depends on dimensions (geometry and shape) of coil, intrinsic properties of material of coil and medium on which coil is wound.
→SI unit of inductance : Henry (H)
→Dimensions : $M ^1 L^2 T^{-2} A^{-2}$
→Other Units :
Wb. $A ^{-1}$, T. $m^2 A^{-1}, N . m . A ^{-2}, \Omega sec$,
V.sec. $A ^{-1}$, V.C. $A ^{-2}, J.A^{-2}$
(i) by changing electric current passing through coil itself
OR
(ii) by changing electric current passing through another coil placed nearby.
→In above both cases, magnetic flux $\phi_{ B }$ linked with coil is directly proportional to electric current I .
$\therefore \phi_{ B } \propto I$
→If dimensions of coil do not change with time, then
$\frac{d \phi_{ B }}{d t} \propto \frac{d I }{d t}$
→If coil has N closely wound turns, then total magnetic flux linked will be $N \phi_{ B }$ which is
also indirectly proportional to electric current passing through coil.
$N \phi_{ B } \alpha I$
The constant of proportionality in equation (1) & (3) is called Inductance.
→Inductance is a scalar quantity. It depends on dimensions (geometry and shape) of coil, intrinsic properties of material of coil and medium on which coil is wound.
→SI unit of inductance : Henry (H)
→Dimensions : $M ^1 L^2 T^{-2} A^{-2}$
→Other Units :
Wb. $A ^{-1}$, T. $m^2 A^{-1}, N . m . A ^{-2}, \Omega sec$,
V.sec. $A ^{-1}$, V.C. $A ^{-2}, J.A^{-2}$
Question 212 Marks
By which different ways can magnetic flux linked with a conducting coil be changed ?
Answer
View full question & answer→→Magnetic flux linked with coil,
$\begin{aligned}
\phi_{ B } & = N (\overrightarrow{ B } \cdot \overrightarrow{ A }) \\
& = NBA \cos \theta \\
& = NBA \cos \omega t
\end{aligned}$
→From above equation, it is clear that by making change in any one or more terms out of $N , B$, $\dot{A}$ and $\theta$, magnetic flux can be changed.
→Thus, magnetic flux can be changed -
→ By rotating coil in magnetic field, changing angular speed $(\omega / \theta)$
→ By changing dimensions of conductor (A)
→ By placing coil in magnetic field changing with time. (B)
→By changing number of turns of the coil $( N )$