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Magnetism And Matter — Physics STD 12 Science — Question

Rajasthan 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 $\text{(weber ampere^{–1})/(volt second ampere^{–1})}$

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:
$\Phi_{1}\propto\text{I}_{2}$
$ \Rightarrow \Phi_{1} = \text{MI}_{2}$
$\therefore\frac{\text{d}\Phi_{1}}{\text{dt}}= \text{M}\frac{\text{dI}_{2}}{\text{dt}}$
$\Rightarrow \text{e} = - \text{M}\frac{\text{dI}_{2}}{\text{dt}}$.

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