Question

The power required to push the arm of a rectangular conductor with a constant speed \(v\) in a motor is directly proportional to

• A. \(v\)
• B. \(v^2\)
• C. \(\sqrt{v}\)
• D. \(\sqrt[3]{v}\)
• E. \(v^3\)

Hint:

In motional emf problems, power often depends on velocity squared because both induced current and force depend on velocity.

Answer 1

The Correct Option is B

Step 1: Recall the concept of motional emf.
When a conductor moves in a magnetic field, an emf is induced: \[ \varepsilon = B l v \] where \(B\) is magnetic field, \(l\) is length, and \(v\) is velocity.

Step 2: Determine the induced current.
If the circuit has resistance \(R\), then current is: \[ I = \frac{\varepsilon}{R} = \frac{Blv}{R} \]

Step 3: Find the magnetic force on the conductor.
The force acting on the conductor is: \[ F = B l I \] Substitute \(I\): \[ F = B l \cdot \frac{Blv}{R} = \frac{B^2 l^2 v}{R} \]

Step 4: Write the expression for power.
Power required is: \[ P = F \cdot v \]

Step 5: Substitute the force expression.
\[ P = \frac{B^2 l^2 v}{R} \cdot v \] \[ P = \frac{B^2 l^2}{R} v^2 \]

Step 6: Identify proportionality.
\[ P \propto v^2 \]

Step 7: Final answer.
\[ \boxed{v^2} \] which matches option \((2)\).