Question

A rectangular wire loop of sides 8 cm and 3 cm with a small cut, is moving out of a region of uniform magnetic field of magnitude 0.3 T directed normal to the plane of the loop. The emf developed across the cut, if the velocity of the loop is 2 cm s⁻¹, in a direction normal to the shorter side of the loop, will be: ____.

• A. 4.8 $\times$ 10⁻⁴ volt
• B. 1.3 $\times$ 10⁻⁴ volt
• C. 1.2 $\times$ 10⁻⁴ volt
• D. 1.8 $\times$ 10⁻⁴ volt

Hint:

If velocity is normal to the short side, it means the loop is moving along the direction of the short side, so the long side acts as the "cutting" length $L$.

Answer 1

The Correct Option is A

Step 1: Understanding the Topic:
This problem is related to "Electromagnetic Induction." It specifically explores the concept of "Motional EMF." When a conductor moves through a magnetic field such that it cuts the magnetic field lines, a potential difference (emf) is induced across its ends.
Step 2: Key Formulas and Approach:
The motional emf ($e$) induced in a straight conductor of length $L$ moving with velocity $v$ in a magnetic field $B$ (where all three are mutually perpendicular) is: \[ e = B \cdot L \cdot v \] In a moving loop, only the segments perpendicular to the motion and passing through the field boundary contribute to the net emf.
Step 3: Detailed Explanation:

Identify given values: $B = 0.3 \text{ T}$ and $v = 2 \text{ cm/s} = 0.02 \text{ m/s}$. The dimensions are $8 \text{ cm}$ and $3 \text{ cm}$.
Determine the "effective" length ($L$): The problem states that the velocity is "normal to the shorter side." This means the loop is moving in the direction of the $3 \text{ cm}$ side.
Consequently, the side that is perpendicular to the velocity is the longer side ($8 \text{ cm}$). This $8 \text{ cm}$ segment is the one "cutting" the field lines as it exits the region.
$L = 8 \text{ cm} = 0.08 \text{ m}$.
Calculate the induced EMF: \[ e = B \times L \times v \] \[ e = 0.3 \text{ T} \times 0.08 \text{ m} \times 0.02 \text{ m/s} \] \[ e = 0.3 \times 0.0016 = 0.00048 \text{ Volts} \]
Format in scientific notation: \[ e = 4.8 \times 10^{-4} \text{ V} \]
Step 4: Final Answer:
The emf developed across the cut is 4.8 $\times$ 10⁻⁴ volt.