An ionized gas contains both positive and negative ions. If it is subjected simultaneously to an ele

Q: Why do we assume the ions are moving? The question doesn't explicitly state they have an initial velocity.

The question mentions the gas is subjected to an electric field. This electric field will accelerate the ions from rest, giving them a velocity parallel or anti-parallel to the electric field.

Q: Why don't the positive and negative ions deflect in opposite directions?

Because the negative ion has both a negative charge AND a negative velocity relative to the positive ion. These two negative signs cancel out in the cross product, resulting in a force in the same direction.

Q: Does the mass of the ions affect the direction of deflection?

No, the mass only affects the magnitude of the acceleration and the radius of curvature of their path, but the direction of the magnetic force is purely determined by the charge, velocity direction, and magnetic field direction.

Q: What happens to the electric force? Does it disappear?

The electric force continues to act along the x-axis, causing the ions to accelerate in that direction. The magnetic force acts along the y-axis, causing them to curve. The net path is a cycloid-like trajectory, but the initial deflection is purely along the y-axis.

Q: Can we use Fleming's Left-Hand Rule here?

Absolutely! For the positive ion, point your index finger along $+z$ (magnetic field) and middle finger along $+x$ (current/velocity). Your thumb points to $-y$. For the negative ion, the velocity is $-x$, but since it's a negative charge, the equivalent current is still along $+x$. So the thumb again points to $-y$!

Question

An ionized gas contains both positive and negative ions. If it is subjected simultaneously to an electric field along the +x-direction and a magnetic field along the +z-direction, then

Options:

(A) positive ions deflect towards +y-direction and negative ions towards –y-direction

(B) all ions deflect towards +y-direction

(C) all ions deflect towards –y-direction

(D) positive ions deflect towards –y-direction and negative ions towards +y-direction

Sigma Percentile · Accepted Answer

Have you ever wondered what happens when you subject a chaotic, ionized gas to both an electric and a magnetic field simultaneously? It sounds like a recipe for absolute pandemonium. You might expect the positive and negative ions to scatter in completely opposite directions, tearing the gas apart.

But physics has a beautiful way of surprising us with hidden symmetries. Let's dive into this classic problem and uncover the elegant dance of charges in crossed fields.

### Setting the Stage: The Crossed Fields

Imagine a region of space where we have set up two uniform fields. We have an electric field $\mathbf{E}$ pointing straight along the positive x-direction, and a magnetic field $\mathbf{B}$ pointing straight up along the positive z-direction.

Mathematically, we can write these as:
$$ \mathbf{E} = E\hat{\mathbf{i}} $$
$$ \mathbf{B} = B\hat{\mathbf{k}} $$

Into this arena, we introduce an ionized gas. This gas is a soup of positively charged ions ($+q$) and negatively charged ions ($-q$). Initially, let's assume they are just milling about. What happens the moment we turn on the fields?

### The Electric Push

The electric field is the first to act. It's the brute force of the electromagnetic world. It doesn't care if you are moving or standing still; if you have a charge, it will push you.

For the positive ions, the electric force $\mathbf{F}_e = q\mathbf{E}$ pushes them in the direction of the field. They accelerate and gain a velocity along the positive x-axis:
$$ \mathbf{v}_+ = v_+\hat{\mathbf{i}} $$

For the negative ions, the story is reversed. The electric force $\mathbf{F}_e = -q\mathbf{E}$ pushes them against the field. They accelerate and gain a velocity along the negative x-axis:
$$ \mathbf{v}_- = -v_-\hat{\mathbf{i}} $$

So far, everything is intuitive. The positive and negative ions are moving in exactly opposite directions.

### The Magnetic Twist

Now comes the twist. As soon as these ions start moving, they awaken the sleeping giant: the magnetic field. The magnetic Lorentz force only acts on moving charges, and its direction is governed by the cross product:
$$ \mathbf{F}_m = q(\mathbf{v} 	imes \mathbf{B}) $$

Let's analyze the positive ion first. It's moving along $\hat{\mathbf{i}}$ and the magnetic field is along $\hat{\mathbf{k}}$. 
$$ \mathbf{F}_{m+} = (+q)(v_+\hat{\mathbf{i}} 	imes B\hat{\mathbf{k}}) $$

Using our right-hand rule, or the cyclic property of unit vectors, we know that $\hat{\mathbf{i}} 	imes \hat{\mathbf{k}} = -\hat{\mathbf{j}}$. 
$$ \mathbf{F}_{m+} = -qv_+B\hat{\mathbf{j}} $$

The positive ion is deflected towards the negative y-direction.

### The Beautiful Cancellation

Now, what about the negative ion? It has a negative charge, and it's moving in the negative x-direction. Let's plug it into the Lorentz force equation:
$$ \mathbf{F}_{m-} = (-q)(-v_-\hat{\mathbf{i}} 	imes B\hat{\mathbf{k}}) $$

Look closely at the math. We have a negative sign from the charge, and a negative sign from the velocity. When we multiply them, the two negatives cancel out perfectly!
$$ \mathbf{F}_{m-} = qv_-B(\hat{\mathbf{i}} 	imes \hat{\mathbf{k}}) $$
$$ \mathbf{F}_{m-} = -qv_-B\hat{\mathbf{j}} $$

The negative ion is *also* deflected towards the negative y-direction!

### The Grand Conclusion

This is the profound beauty of the Lorentz force. Because the negative ion has both an opposite charge AND an opposite velocity compared to the positive ion, the two "opposites" cancel out in the cross product.

The result? The entire ionized gas—both positive and negative ions—is swept together in the exact same direction, deflecting towards the $-y$ axis. It's a stunning example of how mathematical structure dictates physical reality, turning expected chaos into synchronized motion.

Animated Solution for Physics - Magnetic Effects of Current: An ionized gas contains both positive and negative ions. If it is subjected simultaneously to an electric field along the +x-direction and a magnetic field along the +z-direction, then

Select Answer:

Visualized Solution

Coordinate System and Fields

Ionized Gas

Effect of Electric Field

Magnetic Lorentz Force

Deflection of Positive Ion

Deflection of Negative Ion

Conclusion

The Sigma Insight: Motion of a Charge in Magnetic Fields

Setting the Stage

The Electric Push

The Magnetic Twist

The Beautiful Cancellation

The Grand Conclusion

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