Question

Two similar wires of equal lengths are bent in the form of a square and a circular loop. They are suspended in a uniform magnetic field and same current is passed through them. Torque experienced by}

• A. circular loop is greater.
• B. square loop is greater.
• C. both loops is same.
• D. both will be zero.

Hint:

Circle has the maximum area for a fixed perimeter. Torque $\propto$ Area.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A current-carrying loop placed in a uniform magnetic field experiences a magnetic torque.
The magnitude of this torque depends on the current, the area of the loop, the magnetic field strength, and the orientation of the loop.
Since the wires have equal lengths, their perimeters are the same, but the areas they enclose will be different.
Step 2: Key Formula or Approach:
The maximum torque $\tau$ experienced by a current loop of area $A$ carrying current $I$ in a magnetic field $B$ is given by: \[ \tau_{\text{max}} = I \cdot A \cdot B \] Since $I$ and $B$ are the same for both loops, the torque is directly proportional to the enclosed area ($\tau \propto A$).
Step 3: Detailed Explanation:
Let the total length of each wire be $L$.
For the square loop:
The perimeter is $4a = L \implies$ side $a = \frac{L}{4}$.
The area of the square is $A_{\text{square}} = a^2 = \left(\frac{L}{4}\right)^2 = \frac{L^2}{16} = 0.0625 L^2$.
For the circular loop:
The circumference is $2\pi r = L \implies$ radius $r = \frac{L}{2\pi}$.
The area of the circle is $A_{\text{circle}} = \pi r^2 = \pi \left(\frac{L}{2\pi}\right)^2 = \frac{L^2}{4\pi} \approx \frac{L^2}{4 \times 3.14} \approx 0.0796 L^2$.
Comparing the two areas: \[ A_{\text{circle}} = \frac{L^2}{4\pi} \] \[ A_{\text{square}} = \frac{L^2}{16} = \frac{L^2}{4 \times 4} \] Since $\pi \approx 3.14<4$, it follows that $4\pi<16$, and therefore $\frac{1}{4\pi}>\frac{1}{16}$.
So, $A_{\text{circle}}>A_{\text{square}}$.
Because the torque is directly proportional to the area ($\tau = IAB \sin\theta$), the loop with the larger area will experience a greater maximum torque.
Therefore, the circular loop experiences greater torque.
Step 4: Final Answer:
Torque experienced by circular loop is greater.