Animated Solution for Physics - Magnetic Effects of Current: The wire loop PQRSP formed by joining two semicircular wires of radii $R_1$ and $R_2$ carries a current $I$ as shown. The magnitude of the magnetic induction at the centre $C$ is ......

Imagine you are standing at the center $C$ of this intricate wire loop. To understand the total magnetic field you would feel, we must break this complex shape into four distinct, manageable parts: two straight wire segments ($PQ$ and $RS$) and two semicircular arcs ($QR$ and $SP$).

By analyzing the contribution of each segment individually, we can use the principle of superposition to find the net magnetic field.

First, let's look at the straight segments $PQ$ and $RS$. Notice how they lie exactly on the line passing through the center $C$.

According to the Biot-Savart law, the magnetic field produced by a current element is proportional to the cross product of the current vector and the position vector ($d\mathbf{l}\times \mathbf{r}$). Because the point $C$ lies directly on the axis of these wires, the angle between the current and the position vector is either $0^{\circ }$ or $180^{\circ }$.

Since $\sin (0^{\circ })=0$ and $\sin (180^{\circ })=0$, these straight segments produce absolutely zero magnetic field at the center.

Now, the real magic happens with the semicircular arcs. Recall that the magnetic field at the center of a full circular loop is given by the standard formula:

Because our arcs are exactly half of a circle, the magnetic field they produce will be exactly half of this value:

Let's focus on the inner semicircle $QR$. The current here flows in the counter-clockwise direction. If you apply the right-hand thumb rule—curling the fingers of your right hand along the direction of the current—your thumb will point straight out of the screen. This gives us an outward magnetic field $B_1$:

Next, look at the outer semicircle $SP$. Here, the current flows in the clockwise direction. Applying the right-hand rule again, your thumb now points into the screen. This gives us an inward magnetic field $B_2$:

We now have two magnetic fields at the center $C$: $B_1$ pointing outwards and $B_2$ pointing inwards. Because they are in exactly opposite directions, they oppose each other, and the net magnetic field is their difference.

But which one wins? The magnetic field is inversely proportional to the radius. Since the inner semicircle has a smaller radius ($R_1<R_2$), it produces a stronger magnetic field ($B_1>B_2$). Therefore, the net magnetic field will be directed outwards.

Substituting our expressions, we get the final elegant result:

The key takeaway: Whenever you face a complex geometry, break it down into standard shapes. Evaluate the magnitude and direction for each piece, and simply add them up as vectors!