Animated Solution for Physics - Magnetic Effects of Current: A particle of mass $m$ and charge $q$ has an initial velocity $\mathbf{v}=v_0\hat{\mathbf{j}}$. If an electric field $\mathbf{E}=E_0\hat{\mathbf{i}}$ and magnetic field $\mathbf{B}=B_0\hat{\mathbf{i}}$ act on the particle, its speed will double after a time

Imagine you are observing a charged particle suspended at the origin of a 3D coordinate system. Suddenly, it is given an initial velocity ${\mathbf{v}}_0=v_0\hat{\mathbf{j}}$ along the y-axis. At that exact moment, two powerful fields are switched on: an electric field $\mathbf{E}=E_0\hat{\mathbf{i}}$ and a magnetic field $\mathbf{B}=B_0\hat{\mathbf{i}}$, both pointing strictly along the x-axis. Our mission is to find out exactly when the particle's speed doubles.

Before diving into equations, we must understand the distinct roles of these two fields. The magnetic force is given by the Lorentz equation ${\mathbf{F}}_m=q(\mathbf{v}\times \mathbf{B})$. Because of the cross product, this force is always perpendicular to the particle's velocity.

What does this mean physically? A force perpendicular to motion can only change the direction of the particle, never its speed. It does zero work. Therefore, the magnetic field will cause the particle to spiral in the y-z plane, but it will not contribute a single joule to its kinetic energy.

On the other hand, the electric field $\mathbf{E}$ exerts a force ${\mathbf{F}}_e=q\mathbf{E}$ directly along the x-axis. This force provides a constant acceleration in the x-direction.

Using Newton's second law, we can write this acceleration as:

This is the only factor responsible for increasing the particle's speed.

Let's construct the velocity vector at any given time $t$. The particle retains its initial velocity component in the y-z plane (its magnitude remains $v_0$ despite the rotation), and it gains a new velocity component along the x-axis due to the electric acceleration.

To find the speed, we simply calculate the magnitude of this velocity vector using the Pythagorean theorem:

Notice how elegantly the magnetic field's effect is bypassed when we only care about the speed!

The problem asks for the specific time $t$ when the speed becomes exactly twice the initial speed, meaning $|\mathbf{v}(t)|=2v_0$. Let's set up the equation:

To solve this, we square both sides to eliminate the square root. Be careful not to make a silly mistake here—squaring $2v_0$ gives $4v_0^2$:

Subtracting $v_0^2$ from both sides leaves us with:

Taking the square root once more and isolating $t$, we arrive at our final, elegant answer:

This beautiful result shows that the time required depends on the mass, initial velocity, charge, and the strength of the electric field, completely independent of the magnetic field's strength. The resulting path is a helix with an ever-increasing pitch—a true classic in the world of physics!