1. Physical setup

A rectangular conducting loop has one arm $PQ$ moving along the $x$-axis.

• The magnetic field exists only in the region $0 \le x \le b$.
• The rod moves outward from $x=0$ to $x=2b$, and then returns inward from $x=2b$ to $x=0$.

We must identify which graph corresponds to:

• magnetic flux $\Phi$
• induced emf $\mathcal E$
• power dissipated $P$

2. Magnetic flux variation

Let the separation of the moving arm from the fixed side be $x$.

Flux through the loop is

Since field exists only up to $x=b$:

• For $0 \le x \le b$: the area inside field increases linearly with $x$. Hence, $$\Phi \propto x$$
• For $b \le x \le 2b$: the entire field region is already covered, so flux is constant. $$\Phi = \text{constant}$$
• During return from $2b$ to $b$: flux remains constant.
• During return from $b$ to 0: flux decreases linearly back to zero.

So the flux graph is: rises linearly from $0$ to $b$, then stays constant till $2b$, and on return remains constant till $b$, then decreases linearly to zero.

3. Induced emf variation

Induced emf is

Since the rod moves with constant speed, $x$ changes uniformly with time, so emf depends on slope of $\Phi$ vs $x$.

• For $0 \le x \le b$ while moving outward: flux increases linearly, so $$\mathcal E = \text{constant}$$
• For $b \le x \le 2b$ outward: flux constant, so $$\mathcal E = 0$$
• For $2b \to b$ inward: flux still constant, so $$\mathcal E = 0$$
• For $b \to 0$ inward: flux decreases linearly, so emf has constant magnitude again, but opposite sign to the outward case.

Thus emf graph is: constant nonzero from $0$ to $b$, zero from $b$ to $2b$, zero again on return till $b$, then constant nonzero of opposite sign from $b$ to $0$.

So emf must be a rectangular pulse with sign reversal between outward and inward motion.

4. Power dissipated variation

Power dissipated in the loop is

Since power depends on $\mathcal E^2$:

• whenever $\mathcal E$ is nonzero constant, $P$ is a positive constant
• whenever $\mathcal E = 0$, $P=0$

Therefore:

• from $0$ to $b$ outward: constant positive power
• from $b$ to $2b$: zero
• from $2b$ to $b$ inward: zero
• from $b$ to $0$ inward: constant positive power again

So the power graph has two identical positive rectangular pulses, one during entering and one during leaving the field region.

5. Matching with the graphs

From the qualitative shapes:

• Flux: linear rise, then constant plateau, then linear fall on return
• EMF: constant positive, then zero, then constant negative on return
• Power: always positive pulses where emf exists

Hence the correct matching is:

• $A$ = Flux
• $B$ = EMF
• $C$ = Power dissipated

This corresponds to Option B.

6. Comparison with stored answer

Stored correct answer: B

My derived answer: B

So the derived answer agrees with the stored answer.