Animated Solution for Physics - Magnetic Effects of Current: A proton moving with a constant velocity passes through a region of space without any change in its velocity. If $E$ and $B$ represent the electric and magnetic fields respectively. Then, this region of space may have

Mastering the Lorentz Force

Imagine a proton cruising through space, completely undisturbed. Its velocity is perfectly constant—no speeding up, no slowing down, and absolutely no turning. What does this tell us about the forces acting on it?

According to Newton's First Law of Motion, an object will maintain a constant velocity only if the net force acting on it is exactly zero. In the realm of electromagnetism, the total force on a moving charge is governed by the majestic Lorentz Force Equation:

Here, $q$ is the charge of the proton, $\mathbf{E}$ is the electric field, $\mathbf{v}$ is its velocity, and $\mathbf{B}$ is the magnetic field. For the proton to maintain its constant velocity, this entire expression must equal zero. Let's explore the different cosmic scenarios that could make this happen.

The Empty Void

What if the region of space is completely empty of any fields?

If $\mathbf{E}=0$ and $\mathbf{B}=0$, then both the electric force ($q\mathbf{E}$) and the magnetic force ($q(\mathbf{v}\times \mathbf{B})$) vanish instantly. The net force is zero, and our proton sails through smoothly.

This confirms that Option (a) is a perfectly valid scenario.

The Parallel Magnetic Highway

Now, let's introduce a magnetic field, but keep the electric field turned off ($\mathbf{E} = 0, \mathbf{B} eq 0$).

The electric force is zero, but what about the magnetic force? The magnetic force relies on the cross product $\mathbf{v}\times \mathbf{B}$, which has a magnitude of $qvB\sin \theta$. The angle $\theta$ is the angle between the velocity and the magnetic field.

If the proton happens to be moving exactly parallel ($\theta =0^{\circ }$) or anti-parallel ($\theta =180^{\circ }$) to the magnetic field lines, then $\sin \theta =0$. The cross product collapses to zero, and the magnetic force disappears! Even though a magnetic field is present, it exerts no force on the proton.

Thus, Option (b) is also a correct possibility.

The Unbalanced Electric Push

What if we turn on the electric field but turn off the magnetic field ($\mathbf{E} eq 0, \mathbf{B} = 0$)?

The magnetic force is zero, but the electric force ${\mathbf{F}}_e=q\mathbf{E}$ is now active. Because the electric field is non-zero, this force will push the proton, causing it to accelerate. A constant velocity is impossible in this scenario.

Therefore, Option (c) is incorrect.

The Crossed Fields Balancing Act

Finally, the most fascinating scenario: what if both fields are active ($\mathbf{E} eq 0, \mathbf{B} eq 0$)? Can the net force still be zero?

Yes, it can! This requires a delicate balancing act. The electric force ${\mathbf{F}}_e=q\mathbf{E}$ must perfectly cancel out the magnetic force ${\mathbf{F}}_m=q(\mathbf{v}\times \mathbf{B})$. Mathematically, this means:

Imagine the electric field pushing the proton upwards. If we set up a magnetic field pointing into the page, the right-hand rule dictates that the magnetic force will push the proton downwards. If we tune the strengths of these fields just right, the upward push perfectly matches the downward pull. The net force becomes zero, and the proton flies straight through!

This brilliant setup is known as a Velocity Selector, a crucial component in mass spectrometers.

This proves that Option (d) is also a correct scenario.

By systematically applying the Lorentz force equation, we've discovered that a proton can maintain a constant velocity in a complete vacuum, along a parallel magnetic field, or through perfectly balanced crossed fields. The correct options are indeed (a), (b), and (d).