Animated Solution for Physics - Magnetic Effects of Current: A uniform electric field and a uniform magnetic field are acting along the same direction in a certain region. If an electron is projected along the direction of the fields with a certain velocity, then

A Tale of Two Fields

Imagine a region of space where both an electric field $\vec{E}$ and a magnetic field $\vec{B}$ are perfectly aligned, pointing in the exact same direction. Into this highly structured environment, we fire an electron. We shoot it straight along the field lines, meaning its velocity vector $\vec{v}$ is parallel to both $\vec{E}$ and $\vec{B}$.

To understand the fate of this electron, we must analyze the forces acting upon it. Will it curve? Will it speed up? Will it slow down? The answer lies in the fundamental laws of electromagnetism.

Lorentz Force

Whenever a charged particle moves through a region containing both electric and magnetic fields, the total force it experiences is governed by the Lorentz force equation:

This elegant equation is actually a combination of two distinct forces: the electric force ${\vec{F}}_e=q\vec{E}$ and the magnetic force ${\vec{F}}_m=q(\vec{v}\times \vec{B})$. Let's break them down one by one to see how they influence our electron.

A Cross Product Mystery

Let's look at the magnetic component first. The magnetic force depends on the cross product of the velocity vector and the magnetic field vector. The magnitude of this force is given by:

where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$. In our specific scenario, the electron is projected along the direction of the magnetic field. This means the angle $\theta$ is exactly $0^{\circ }$.

Since $\sin (0^{\circ })=0$, the entire cross product collapses to zero. The magnetic field, despite being present, exerts absolutely no force on the electron. The electron is completely blind to the magnetic field as long as it moves perfectly parallel to it.

The Power of a Negative Charge

Now, let's turn our attention to the electric force. The electric force is much simpler; it doesn't care about the particle's velocity or direction of motion. It only cares about the charge and the electric field:

Here is where the crucial detail comes in: our particle is an electron. An electron carries a negative charge, so $q=-e$. Substituting this into our equation gives:

That negative sign is everything. It tells us that the electric force acts in the exact opposite direction of the electric field. Since the electron was moving in the direction of the electric field, the force is pushing against it, directly opposing its motion.

Deceleration

We have established that the magnetic force is zero and the electric force is pushing backward against the electron's velocity.

When a net force acts in the direction opposite to an object's motion, it causes deceleration. The electron will continue to move in a straight line (because there are no sideways forces to turn it), but it will progressively lose speed. Therefore, the correct conclusion is that its velocity will decrease.