CBSE Class 12 Physics: Solution & Marking Scheme

Given Data:

• Current in long wire ($I_1$) = $2\text{ A}$
• Current in rectangular loop ($I_2$) = $1\text{ A}$
• Let the loop dimensions be length $l$ (parallel to wire) and width $b$ (perpendicular to wire).

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(i) Torque acting on the loop: [1 Mark]

Step 1 (Governing Principle): [0.5 Mark]

The magnetic field produced by the long wire is perpendicular to the plane of the loop at all points. The force on each segment of the loop lies in the same plane as the loop. Since the force vectors are coplanar with the loop and act along the edges, the net torque $\vec{\tau} = \vec{r} \times \vec{F}$ is zero.

Step 2 (Conclusion): [0.5 Mark]

Since the magnetic force on each side of the loop acts in the plane of the loop, the line of action of these forces passes through the center (or the forces are balanced such that they do not produce rotation). Thus, Torque $\tau = 0$.

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(ii) Magnitude and direction of the net force: [2 Marks]

Step 1 (Formula & Setup): [0.5 Mark]

The magnetic field at a distance $r$ from the long wire is $B = \frac{\mu_0 I_1}{2\pi r}$. The force on a current-carrying segment is $d\vec{F} = I_2(d\vec{l} \times \vec{B})$.

Step 2 (Mathematical Working): [1 Mark]

The forces on the two sides perpendicular to the wire are equal and opposite, hence they cancel out. We only consider the two sides parallel to the wire at distances $r_1$ and $r_2$:

$$F_{net} = \left| \frac{\mu_0 I_1 I_2 l}{2\pi r_1} - \frac{\mu_0 I_1 I_2 l}{2\pi r_2} \right|$$

Substituting the values (assuming standard dimensions $r_1, r_2$ and $l$):

$$F_{net} = \frac{\mu_0 I_1 I_2 l}{2\pi} \left( \frac{1}{r_1} - \frac{1}{r_2} \right)$$

Step 3 (Final Answer): [0.5 Mark]

The magnitude is calculated by substituting the specific dimensions provided in the figure. The direction of the net force is towards the wire if the currents are parallel (attractive) or away from the wire if the currents are anti-parallel (repulsive), determined by Fleming's Left-Hand Rule.

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💡 EL10 Concept Insight:

Think of the loop like a magnet being pulled by the wire; because the wire's magnetic field is stronger closer to it, the side of the loop nearest to the wire gets "pulled" harder than the far side, resulting in a net force. However, because all these "tugs" happen in the same flat plane, the loop just wants to slide toward or away from the wire rather than spin around, which is why the torque is zero.