Question
Obtain an expression for the self-inductance of a solenoid.
✓
Answer
i. Consider a current I established in the windings (turns) of a long solenoid. The current produces a magnetic flux $\phi_{ B }$ through the central region.
ii. The inductance of the solenoid is given by,
$L =\frac{ N \phi_{ B }}{ I } \text {, }$
where $N =$ the number of turns, $\Phi_{ B }=$ magnetic flux linkage.
iii. The flux linkage for a length I near the middle of the solenoid is,
$N \Phi_{ B }=( nl )(\overrightarrow{ B } \cdot \overrightarrow{ A })= nlBA ,\left(\text { for } \theta=0^{\circ}\right. \text { ), }$
where $n =$ the number of turns per unit length, $B=$ magnetic field
$A=$ the cross-sectional area of the solenoid.
iv. The magnetic field inside the solenoid is given as, $B=$ $\mu_0 ni$
v. Hence, $L=\frac{N \phi_B}{i}$
$ =\frac{( nl ) BA }{ i }$
$=\frac{ nl \left(\mu_0 ni \right) A }{ i }$
$\therefore L =\mu_0 n ^2 A $
where, $Al$ is the interior volume of solenoid.
ii. The inductance of the solenoid is given by,
$L =\frac{ N \phi_{ B }}{ I } \text {, }$
where $N =$ the number of turns, $\Phi_{ B }=$ magnetic flux linkage.
iii. The flux linkage for a length I near the middle of the solenoid is,
$N \Phi_{ B }=( nl )(\overrightarrow{ B } \cdot \overrightarrow{ A })= nlBA ,\left(\text { for } \theta=0^{\circ}\right. \text { ), }$
where $n =$ the number of turns per unit length, $B=$ magnetic field
$A=$ the cross-sectional area of the solenoid.
iv. The magnetic field inside the solenoid is given as, $B=$ $\mu_0 ni$
v. Hence, $L=\frac{N \phi_B}{i}$
$ =\frac{( nl ) BA }{ i }$
$=\frac{ nl \left(\mu_0 ni \right) A }{ i }$
$\therefore L =\mu_0 n ^2 A $
where, $Al$ is the interior volume of solenoid.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
An electron in an atom revolves around the nucleus in an orbit of radius $0.53 \mathring A $. If the frequency of revolution of an electron is $9 \times 10^9 MHz$, calculate the equivalent magnetic moment. $\left[e=1.6 \times 10^{-19} C \right]$
→Prove theoretically, the relation between e.m.f. induced and rate of change of magnetic flux in a coil moving in a uniform magnetic field.
Obtain an expression for torque acting on a body rotating with uniform angular acceleration.
Explain the working of transistor as a switch.
For diver's safety, a $10 \mathrm{~m}$ platform diving pool should be $5 \mathrm{~m}$ deep. However, with an excellent dive, a diver usually reaches a maximum depth of $2.5 \mathrm{~m}$.
(i) Calculate the pressure due to the weight of the water at the depth of $2.5 \mathrm{~m}$.
(ii) Calculate the depth below the surface of water at which the pressure due to the weight of the water equals $1.0 \mathrm{~atm}$. [Density of water $=10^3 \mathrm{~kg} / \mathrm{m}^3, 1 \mathrm{~atm}=101.3 \mathrm{kPa}$ ]
→(i) Calculate the pressure due to the weight of the water at the depth of $2.5 \mathrm{~m}$.
(ii) Calculate the depth below the surface of water at which the pressure due to the weight of the water equals $1.0 \mathrm{~atm}$. [Density of water $=10^3 \mathrm{~kg} / \mathrm{m}^3, 1 \mathrm{~atm}=101.3 \mathrm{kPa}$ ]
A body of surface area $10 \mathrm{~cm}^2$ and temperature $727^{\circ} \mathrm{C}$ emits $300 \mathrm{~J}$ of energy per minute.
Find its emissivity.
→Find its emissivity.
A particle of charge q and momentum p enters a region of uniform magnetic field B travelling at right angles to the field, and is deflected through a right angle as shown. Obtain an expression for the length of the particle’s path In terms of q, p and B.
A ray of light is incident on a water surface of refractive index $\frac{4}{3}$ making an angle of $40^{\circ}$ with the surface. Find the angle of refraction.
→Draw a diagram showing the linear velocity, angular velocity and radial acceleration of a particle performing circular motion with radius r.
In an adiabatic compression of a gas with $\gamma=1.4$, the final pressure is double the initial pressure. If the initial temperature of the gas is $300 K$, find the final temperature.
→