Question

A square loop of side length $a$ and resistance $R$ is moved in the region of uniform magnetic field $B$ by external agent (loop remains completely inside field throughout the motion), with a constant velocity $v$ through a distance $x$. The work done by external

• A. $\frac{B^{2} a^{3} v}{R}$
• B. Zero
• C. $\frac{B^{2} a^{2} v}{R}$
• D. $\frac{B a^{2} v}{R}$

Answer 1

Answer: (B)

Zero

Answer 2

Explanation:

1. Magnetic Flux: Since the square loop remains completely inside the uniform magnetic field $B$ throughout the motion, the magnetic flux ($\Phi$) through the loop remains constant. The area $A = a^2$ is constant, so $\Phi = B \cdot A = B a^2$ is constant.

2. Induced EMF: According to Faraday's Law of Electromagnetic Induction, the induced electromotive force (EMF) ($\mathcal{E}$) is given by:

   $$\mathcal{E} = -\frac{d\Phi}{dt}$$

   Since the magnetic flux $\Phi$ is constant, its time derivative $\frac{d\Phi}{dt} = 0$. Therefore, the induced EMF $\mathcal{E} = 0$.

3. Induced Current: The induced current ($I_{ind}$) in the loop is given by Ohm's Law:

   $$I_{ind} = \frac{\mathcal{E}}{R}$$

   Since $\mathcal{E} = 0$, the induced current $I_{ind} = \frac{0}{R} = 0$.

4. Magnetic Force: A magnetic force acts on a current-carrying conductor in a magnetic field. Since there is no induced current ($I_{ind} = 0$) in the loop, there is no magnetic force exerted on the loop.

5. Work Done by External Agent: To move the loop at a constant velocity $v$, the net force on the loop must be zero. Since there is no opposing magnetic force, the external force ($F_{ext}$) required to maintain constant velocity is zero. The work done ($W$) by the external agent is given by $W = F_{ext} \cdot (\text{distance})$. Since $F_{ext} = 0$, the work done by the external agent is $W = 0 \cdot x = 0$.

Therefore, no work is done by the external agent.