Animated Solution for Physics - Magnetic Effects of Current: A metallic block carrying current $I$ is subjected to a uniform magnetic induction $B$ as shown in figure. The moving charges experience a force $F$ given by ......which results in the lowering of the potential of the face....... Assume the speed of the charges to be $v$.

Unveiling the Hidden Forces Inside a Wire

Have you ever wondered what happens to the tiny electrons zipping through a wire when you bring a magnet nearby? They don't just ignore it; they experience a fundamental force of nature that pushes them sideways. This beautiful interplay between electricity and magnetism is exactly what we are going to explore in this problem.

Imagine a solid metallic block. We are told that a current $I$ is flowing through it along the positive X-axis. But here is the first crucial catch: in metals, the actual charge carriers are free electrons, and they carry a negative charge.

Because they are negative, to create a conventional current in the positive X-direction, these electrons must actually be drifting in the opposite direction. Therefore, the velocity vector of our electrons is pointing along the negative X-axis:

Simultaneously, the block is bathed in a uniform magnetic field $\mathbf{B}$ pointing straight up, along the positive Y-axis:

Lorentz Force

To find out how these electrons react to the magnetic field, we invoke the Lorentz force law. The magnetic force ${\mathbf{F}}_m$ on a moving charge is given by the cross product of its velocity and the magnetic field, scaled by its charge:

Let's carefully substitute our specific vectors into this master equation. Remember, the charge of an electron is $q=-e$.

Now, it's just a matter of vector algebra. The two negative signs beautifully cancel each other out:

From our knowledge of unit vectors, we know that the cross product of $\hat{i}$ and $\hat{j}$ gives us $\hat{k}$, the unit vector pointing along the positive Z-axis.

This is our first answer! The force experienced by the moving charges is $evB\hat{k}$.

But what does this mean physically? The positive Z-axis points directly outwards from the page, towards the front face of our metallic block. In our diagram, this front face is labeled $ABCD$.

Because the electrons are being constantly pushed towards this face, they will start to accumulate there. Since electrons carry a negative charge, this buildup of negative charge will naturally lower the electric potential of face $ABCD$.

This fascinating phenomenon, where a magnetic field pushes charge carriers to one side of a conductor creating a voltage difference, is known as the Hall Effect. It is a powerful tool used by physicists to determine the sign and density of charge carriers in various materials!