Animated Solution for Physics - Magnetic Effects of Current: An infinitely long conductor $PQR$ is bent to form a right angle as shown in figure. A current $I$ flows through $PQR$. The magnetic field due to this current at the point $M$ is $H_1$. Now, another infinitely long straight conductor $QS$ is connected at $Q$, so that current is $I/2$ in $QR$ as well as in $QS$, the current in $PQ$ remaining unchanged. The magnetic field at $M$ is now $H_2$. The ratio $H_1/H_2$ is given by

The problem of finding the magnetic field at a specific point due to a complex wire geometry is a classic in physics. It tests your understanding of the Biot-Savart Law and the principle of superposition. Let's embark on this thrilling journey to decode the magnetic field at point $M$.

Imagine you are standing at point $M$, looking down at the wire $PQR$. The wire is bent at a perfect right angle at the origin $Q$. A steady current $I$ flows from $-\infty$ through $P$ to $Q$, and then downwards to $R$ towards $-\infty$.

We are given that the total magnetic field at $M$ in this initial configuration is $H_1$. But what exactly contributes to this field? To answer this, we must invoke the fundamental law of magnetostatics.

According to the Biot-Savart Law, the magnetic field $d\vec{B}$ produced by a small current element $d\vec{l}$ is proportional to the cross product $d\vec{l}\times \vec{r}$.

Notice something fascinating about the wire $QR$. Point $M$ lies exactly on the extended axis of this wire! For any current element on $QR$, the position vector $\vec{r}$ pointing towards $M$ is perfectly anti-parallel to the current direction $d\vec{l}$.

Because the cross product of parallel or anti-parallel vectors is zero, the magnetic field contribution from the entire wire $QR$ at point $M$ is absolutely zero!

This is a massive simplification. It means that the initial magnetic field $H_1$ is solely generated by the semi-infinite wire $PQ$.

Adding a New Wire

Now, the problem introduces a twist. A new infinitely long straight conductor $QS$ is connected at $Q$, extending along the positive x-axis.

The current $I$ flowing through $PQ$ reaches the junction $Q$ and splits. The problem states that the current becomes $I/2$ in $QR$ and $I/2$ in $QS$. Crucially, the current in $PQ$ remains unchanged at $I$.

In this new configuration, the total magnetic field at $M$ is $H_2$. By the principle of superposition, $H_2$ is the vector sum of the magnetic fields from all three wire segments: $PQ$, $QR$, and $QS$.

Let's evaluate each component: 1. Wire PQ: The current is still $I$, and its geometry hasn't changed. So, its contribution remains $H_1$. 2. Wire QR: Point $M$ is still on its axis. Even though the current is now $I/2$, the cross product is still zero. So, its contribution remains $0$. 3. Wire QS: This is a new semi-infinite wire. It has the exact same geometric relationship to $M$ as $PQ$ does (both are semi-infinite wires starting from the origin and extending along an axis perpendicular to $M$). However, the current in $QS$ is $I/2$. Since the magnetic field is directly proportional to the current, the field produced by $QS$ will be exactly half of the field produced by $PQ$.

Now, we simply add these contributions together to find the new total magnetic field $H_2$.

The question asks for the ratio of the initial field $H_1$ to the new field $H_2$.

And there we have it! By carefully analyzing the geometry and applying the Biot-Savart law, we've elegantly arrived at the final answer.