A charged particle moves through a magnetic field perpendicular to its direction. Then,
(a) the mome

Q: Why doesn't the magnetic field do any work?

Work is the dot product of force and displacement. Since the magnetic force is always perpendicular to the velocity (and thus displacement), the dot product is zero.

Q: If the force is non-zero, shouldn't the kinetic energy change?

A force only changes kinetic energy if it has a component along the direction of motion. A purely perpendicular force only changes the direction of motion, not the speed.

Q: How can momentum change if kinetic energy is constant?

Kinetic energy depends on the magnitude of velocity (speed), which is constant. Momentum depends on the vector velocity. Since the direction changes, the vector velocity changes, and so does momentum.

Q: What path does the particle take?

Because the force is always perpendicular to the velocity and constant in magnitude, it acts as a centripetal force, causing the particle to move in a uniform circular path.

Q: What if the particle enters at an angle other than 90 degrees?

The velocity component parallel to the field remains unchanged, while the perpendicular component causes circular motion. The resulting path is a helix.

Question

A charged particle moves through a magnetic field perpendicular to its direction. Then,

Options:

(A) the momentum changes but the kinetic energy is constant

(B) both momentum and kinetic energy of the particle are not constant

(C) both momentum and kinetic energy of the particle are constant

(D) kinetic energy changes but the momentum is constant

Sigma Percentile · Accepted Answer

### The Setup: A Dance of Charges and Fields

Imagine you are a tiny, positively charged particle, zooming through space.

Suddenly, you enter a region filled with a uniform magnetic field.

But you don't just enter any which way—you dive in perfectly perpendicular to the magnetic field lines.

What happens next is one of the most elegant dances in physics.

### The Lorentz Force

When a charged particle moves through a magnetic field, it experiences a force.

But this isn't just a simple push or pull.

This is the **Lorentz force**, and it has a very specific, almost quirky rule: it always acts perpendicular to both your velocity and the magnetic field.

Mathematically, we write this as:

$$ \vec{F}_m = q(\vec{v} 	imes \vec{B}) $$

Because of this cross product, the force doesn't push you forward to speed you up, nor does it pull you backward to slow you down.

Instead, it pushes you sideways.

### The Master Equation: Work and Energy

Now, let's ask a crucial question: Does this magnetic force do any work on you?

In physics, work is defined as the dot product of force and displacement:

$$ W = \int \vec{F} \cdot d\vec{s} $$

Since the magnetic force is always perpendicular to your direction of motion (your displacement), the angle $	heta$ between them is exactly $90^\circ$.

And what is the cosine of $90^\circ$? It's zero!

$$ W = \int F ds \cos 90^\circ = 0 $$

The magnetic field does absolutely **zero work** on the charged particle.

### The Conservation of Speed

Why is this so important?

Enter the **Work-Energy Theorem**, which states that the net work done on an object equals its change in kinetic energy:

$$ W = \Delta KE $$

Since $W = 0$, it immediately follows that $\Delta KE = 0$.

Your kinetic energy is perfectly conserved.

And since kinetic energy is given by $KE = \frac{1}{2}mv^2$, your speed $v$ must also remain absolutely constant.

You don't speed up, and you don't slow down.

### The Twist: Momentum and Direction

If your speed is constant, you might be tempted to think your momentum is constant too.

But here is where many students fall into a classic trap!

Kinetic energy is a **scalar** quantity.

It only cares about the magnitude of your velocity (your speed).

Momentum, however, is a **vector** quantity.

It cares about both your speed *and* your direction:

$$ \vec{p} = m\vec{v} $$

Because the magnetic force is constantly pushing you sideways, it forces you into a circular path.

You are constantly turning.

Even though your speedometer reads a constant number, your compass is spinning wildly!

### Final Conclusion

Since your direction of motion is continuously changing, your velocity vector $\vec{v}$ is changing.

And if your velocity vector is changing, your momentum vector $\vec{p}$ must also be changing.

So, we arrive at our beautiful, counter-intuitive conclusion:

The **kinetic energy** of the particle remains perfectly constant because the magnetic force does no work.

However, the **momentum** of the particle is constantly changing because the magnetic force continuously alters the direction of motion.

Animated Solution for Physics - Magnetic Effects of Current: A charged particle moves through a magnetic field perpendicular to its direction. Then,

Select Answer:

Visualized Solution

The Setup $v ⊥ B$

Magnetic Force $F_{m}$

Work Done $W = 0$

Kinetic Energy $K E$

Momentum $p$

Conclusion $Δ K E = 0$

The Way Forward $θ \neq = 9 0^{\circ}$

The Sigma Insight: Motion of a Charge in Magnetic Fields

The Setup

The Lorentz Force

The Master Equation

The Conservation of Speed

The Twist

Final Conclusion

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