Question

A long metal rod of length \(L\) completes the circuit as shown. The area of the circuit is perpendicular to magnetic field \(B\). Total resistance of the circuit is \(R\). The force needed to move the rod in the direction as shown with constant speed \(V\) is

• A. \( \dfrac{B^2LV}{R} \)
• B. \( \dfrac{BLV}{R} \)
• C. \( \dfrac{BLV^2}{R} \)
• D. \( \dfrac{B^2L^2V}{R} \)

Hint:

Mechanical power equals electrical power in electromagnetic induction problems.

Answer 1

The Correct Option is D

Step 1: Induced emf in the rod.
\[ \varepsilon = BLV. \]
Step 2: Induced current.
\[ I = \frac{\varepsilon}{R} = \frac{BLV}{R}. \]
Step 3: Magnetic force on the rod.
\[ F = BIL = B \left(\frac{BLV}{R}\right)L. \]
Step 4: Final expression.
\[ F = \frac{B^2L^2V}{R}. \]
Step 5: Conclusion.
Required force is \( \dfrac{B^2L^2V}{R} \).