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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 c1 and c2 are placed close to each other. Find out the expression for the emf induced in the coil c1 due to a change in the current through the coil c2.
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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 I2 flow through S2.This sets up a magnetic flux $\phi_{1}$ through each turn of the coil S1.

Total flux linked with S1

$\text{N}_{1}\phi_{1} = \text{M}_{12}\text{I}_{2}$......(i)

where M12 is the mutual inductance between the two solenoids Magnetic field due to the current I2 in S2 is $\text{S}_{2}\text{ is } \mu_{\circ}\text{n}_{2}\text{I}_{2}$

Therefore, resulting flux linked with S1.

$\text{N}_{1}\phi_{1} = [ (\text{n}_{1}\ell)\pi\text{r}^{2}](\mu_0\text{n}_{2}\text{I}_{2})$........(ii)

Comparing (i) & (ii),we get

M12 I2 $ =(\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 C1 due to current (I2) in coil C2;

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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