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Physics · 2025 · 3 marks
CBSE 2025 · Region 6 · Set 1 · Q24
(a) State Lenz's law. A rod MN of length L is rotated about an axis passing through its end M perpendicular to its length, with a constant angular…
(a)
State Lenz's law. A rod MN of length L is rotated about an axis passing through its end M perpendicular to its length, with a constant angular velocity $\displaystyle \omega$ in a uniform magnetic field $\displaystyle \overrightarrow{\mathrm{B}}$ parallel to the axis. Obtain an expression for emf induced between its ends.
OR
(b)
Define 'self-inductance' of a coil. Derive an expression for self-inductance of a long solenoid of cross-sectional area A and length $\displaystyle l$, having n turns per unit length.
Marking-scheme solution
Lenz's law
The polarity of induced emf is such that it tends to produce a current which opposes the change in magnetic flux that produced it.
Expression of induced emf
The magnitude of the emf generated across the length dr of the rod as it moves at right angles to the magnetic field is given by
\[d\varepsilon = B\mathrm{v}\,dr \]
\[\varepsilon = \int d\varepsilon \]
\[= \int_0^L B\mathrm{v}\,dr \]
\[\varepsilon = \int_0^L B\omega r\,dr \]
\[\varepsilon = \frac{1}{2} B L^2 \omega \]
Alternatively:
Area of the sector (QMN) $\displaystyle = \dfrac{1}{2} L^2 \theta$
Induced emf is $\displaystyle \varepsilon = B \times \dfrac{d}{dt}\left( \dfrac{1}{2} L^2 \theta \right)$
\[\varepsilon = \frac{1}{2} B L^2 \frac{d\theta}{dt} \]
\[\varepsilon = \frac{1}{2} B L^2 \omega \]
Self inductance of a coil is the ratio of the flux linkage to the current flowing in the coil.
Alternatively:
Self inductance of a coil is defined as the flux linked with the coil when unit current flows through it.
Alternatively:
Self inductance of a coil may be defined as the magnitude of emf induced in the coil when current changes at the rate of $\displaystyle 1$ A/s in the coil.
Expression for self inductance of a long solenoid:
The magnetic field due to current flowing in the solenoid, $\displaystyle B = \mu_0 n I$
Total flux linked with the given solenoid
\[N\phi_B = (nl)(\mu_0 n I)\,A \]
\[N\phi_B = \mu_0 n^2 A\, l I \]
Self inductance
\[L = \frac{N\phi_B}{I} \]
\[L = \mu_0 n^2 A\, l \]
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