Question

If a conducting rod of length $100\text{ cm}$ rotates about one of its ends with a constant frequency of $14\text{ revolutions per second}$ in a plane perpendicular to a uniform magnetic field of $2\text{ T}$, then the induced emf between the two ends of the rod is:

• A. $144\text{ V}$
• B. $88\text{ V}$
• C. $122\text{ V}$
• D. $230\text{ V}$

Hint:

For a rod rotating about one end in a uniform magnetic field, \[ \varepsilon=\frac{1}{2}B\omega L^2 \] or equivalently, \[ \varepsilon=\pi BfL^2 \] Memorize both forms. In MCQs, the second form is often quicker because frequency $f$ is usually given directly instead of angular velocity $\omega$.

Answer 1

The Correct Option is B

Concept: When a conducting rod rotates in a uniform magnetic field about one of its ends, the free charges present in the rod experience a magnetic Lorentz force. This causes charge separation along the length of the rod and an emf is induced between its ends. For a rod of length $L$ rotating with angular velocity $\omega$ in a magnetic field $B$ perpendicular to the plane of rotation, the induced emf is given by \[ \varepsilon = \frac{1}{2}B\omega L^2 \] Since angular velocity and frequency are related by \[ \omega = 2\pi f \] the expression for induced emf can also be written as \[ \varepsilon = \frac{1}{2}B(2\pi f)L^2 = \pi BfL^2 \]

Step 1: Convert the given quantities into SI units Length of the rod: \[ L = 100\text{ cm} = 1\text{ m} \] Frequency of rotation: \[ f = 14\text{ revolutions per second} \] Magnetic field: \[ B = 2\text{ T} \]

Step 2: Calculate the angular velocity Using \[ \omega = 2\pi f \] \[ \omega = 2\pi(14) = 28\pi\ \text{rad s}^{-1} \]

Step 3: Apply the formula for induced emf \[ \varepsilon = \frac{1}{2}B\omega L^2 \] Substituting the values, \[ \varepsilon = \frac{1}{2}\times 2\times 28\pi \times (1)^2 \] \[ \varepsilon = 28\pi \] Using \[ \pi \approx \frac{22}{7} \] \[ \varepsilon = 28\times\frac{22}{7} \] \[ \varepsilon = 4\times22 \] \[ \varepsilon = 88\text{ V} \] Therefore, the induced emf between the two ends of the rod is \[ \boxed{88\text{ V}} \]