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Magnetism And Matter — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsMagnetism And Matter5 Marks
Question
  1. Define mutual inductance and write its S.I. unit.
  2. Derive an expression for the mutual inductance of two long co-axial solenoids of same length wound one over the other.
  3. In an experiment, two coils $c_1 $ and $c_2$ are placed close to each other. Find out the expression for the emf induced in the coil $c_1$ due to a change in the current through the coil $c_2.$
✓

Answer

  1. $\phi = \text{MI}$
Mutual inductance of two coils is equal to the magnetic flux linked with one coil when a unit current is passed in the other coil.
Alternate Answer
$\text{e} = - \text{M}\frac{\text{dl}}{\text{dt}}$
Mutual inductance is equal to the induced emf set up in one coil when the rate of change of current flowing through the other coil is unity.
SI unit: henry/(weber ampere$^{–1})/($volt second ampere$^{–1})$

Let a current $I_2$ flow through $S_2$.This sets up a magnetic flux $\phi_{1}$ through each turn of the coil $S_1.$
Total flux linked with $S_1$
$\text{N}_{1}\phi_{1} = \text{M}_{12}\text{I}_{2}$......(i)
where $M_{12}$ is the mutual inductance between the two solenoids Magnetic field due to the current $I_2$ in $S_2$ is $\text{S}_{2}\text{ is } \mu_{\circ}\text{n}_{2}\text{I}_{2}$
Therefore, resulting flux linked with $S_{1.}$
$\text{N}_{1}\phi_{1} = [ (\text{n}_{1}\ell)\pi\text{r}^{2}](\mu_0\text{n}_{2}\text{I}_{2})$........(ii)
Comparing (i) & (ii),we get
$M_{12}I_2 =(\text{n}_{1}\ell)\pi\text{r}_{1}^{2}(\mu_{0}\text{n}_{2}\text{I}_{2})$
$\therefore\text{M}_{12} = \mu_{0}\text{n}_{1}\text{n}_{2}\pi\text{r}_{1}^{2}\ell$
  1. Let a magnetic flux be $(\phi_{1})$linked with coil $C_1$ due to current (I_2) in coil $C_{2};​​​​​​​$​​​​​​​
We have:
$\Phi_{1}\propto\text{I}_{2}$
$ = > \Phi_{1} = \text{MI}_{2}$
$\therefore\frac{\text{d}\Phi_{1}}{\text{dt}}= \text{M}\frac{\text{dI}_{2}}{\text{dt}}$
$= > \text{e} = - \text{M}\frac{\text{dI}_{2}}{\text{dt}}$.

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