An electron gun is placed inside a long solenoid of radius R on its axis. The solenoid has n turns/l

Q: Why is the maximum distance from the axis $2r$ and not $r$?

The electron starts its motion *from* the axis. Since it moves in a circle, the axis lies on the circumference of this circle. The furthest point on a circle from a point on its circumference is diametrically opposite, which is a distance of $2r$ away.

Q: Why is the magnetic field perpendicular to the velocity?

The problem states the electron is shot 'along the radius'. The magnetic field of a long solenoid is parallel to its central axis. A radial line is always perpendicular to the central axis, hence $\vec{v} \perp \vec{B}$.

Q: What happens if the electron is shot at an angle other than $90^\circ$ to the axis?

If shot at an angle, the electron's velocity would have a component parallel to the magnetic field. This would result in a helical path instead of a simple circular one.

Q: Does the electron's speed change as it moves?

No. The magnetic force is always perpendicular to the velocity, meaning it does no work on the electron. Therefore, the kinetic energy and speed remain constant.

Q: Why do we use $B = \mu_0 n I$ for the magnetic field?

This is the standard formula for the uniform magnetic field inside an ideal, infinitely long solenoid, where $n$ is the number of turns per unit length and $I$ is the current.

Question

An electron gun is placed inside a long solenoid of radius $R$ on its axis. The solenoid has $n$ turns/length and carries a current $I$. The electron gun shoots an electron along the radius of the solenoid with speed $v$. If the electron does not hit the surface of the solenoid, maximum possible value of $v$ is (all symbols have their standard meaning)

Options:

(A) $\frac{2e\mu_0nIR}{m}$

(B) $\frac{e\mu_0nIR}{4m}$

(C) $\frac{e\mu_0nIR}{2m}$

(D) $\frac{e\mu_0nIR}{m}$

Sigma Percentile · Accepted Answer

Welcome, future engineers and physicists! Today, we are going to tackle a fascinating problem from the JEE Main 2020 paper. At first glance, this question might seem like a dense mix of electromagnetism and geometry, but I promise you, once we break it down, it reveals a beautiful and elegant puzzle.

Imagine you are standing right on the central axis of a massive, infinitely long solenoid. You have an electron gun in your hand, and you are about to fire an electron straight outwards, along the radius. Our mission? To find the absolute maximum speed at which we can fire this electron so that it *just* grazes the inner wall of the solenoid without crashing into it.

Let's dive into the physics of this microscopic dance!

### Visualizing the Arena

First, let's set the stage. A long solenoid carries a current $I$ through its $n$ turns per unit length. This creates a uniform magnetic field $\vec{B}$ inside it. The crucial detail here is the direction: the magnetic field inside a solenoid is perfectly parallel to its central axis.

Now, our electron is shot radially outwards. This means its velocity vector $\vec{v}$ is pointing straight away from the axis. Geometrically, a radial line is always perpendicular to the central axis. Therefore, the electron's velocity $\vec{v}$ is exactly perpendicular to the magnetic field $\vec{B}$.

### The Dance of the Electron

What happens when a charged particle moves perpendicular to a magnetic field? It experiences a magnetic Lorentz force. According to the right-hand rule (and remembering that an electron has a negative charge), this force is perpendicular to both the velocity and the magnetic field.

Because this force is always perpendicular to the electron's motion, it does no work. It doesn't speed the electron up or slow it down; it only changes its direction. This is the classic recipe for uniform circular motion! The magnetic force acts as the centripetal force, locking the electron into a circular orbit.

The radius $r$ of this circular path is a well-known formula derived by equating the magnetic force to the centripetal force:

$$r = \frac{mv}{eB}$$

### The Geometric Trap

Here is where many students make a critical mistake. It is tempting to say, "Oh, the electron shouldn't hit the wall, so the radius of its path $r$ must be less than the radius of the solenoid $R$."

**Stop and visualize!**

The electron does *not* start at the center of its circular orbit. It starts its journey from the axis of the solenoid. Since it immediately begins to curve into a circle, the starting point (the solenoid's axis) lies exactly on the *circumference* of the electron's circular path.

If you stand on the edge of a circle, what is the furthest point you can reach on that circle? It's the point diametrically opposite to you! Therefore, the maximum distance the electron will ever reach from the solenoid's axis is the **diameter** of its circular path, which is $2r$.

### The Limiting Condition

To ensure our electron survives its journey and doesn't crash into the solenoid wall (which is at a distance $R$ from the axis), its maximum reach must be less than or equal to $R$.

Mathematically, this constraint is beautifully simple:

$$2r_{\max} = R$$

Solving for the maximum radius of the electron's path, we get:

$$r_{\max} = \frac{R}{2}$$

### Bringing in the Physics

Now that we have cracked the geometry, the physics is a breeze. We take our limiting radius $r_{\max}$ and plug it back into our radius formula:

$$\frac{R}{2} = \frac{mv_{\max}}{eB}$$

We want to find the maximum speed, $v_{\max}$. Let's rearrange the equation to isolate it:

$$v_{\max} = \frac{eBR}{2m}$$

### The Final Piece of the Puzzle

We are almost there! The problem doesn't give us $B$ directly; it gives us the parameters of the solenoid. We know that the magnetic field inside an ideal, long solenoid is given by:

$$B = \mu_0 n I$$

Let's substitute this expression for $B$ into our velocity equation:

$$v_{\max} = \frac{e(\mu_0 n I)R}{2m}$$

Cleaning it up, we arrive at our final, elegant answer:

$$v_{\max} = \frac{e\mu_0 n I R}{2m}$$

And there you have it! By carefully visualizing the geometry of the starting position and combining it with the fundamental laws of electromagnetism, we've successfully navigated this JEE problem. Remember, in physics, a good diagram and a clear mental picture are your most powerful tools. Keep practicing, and stay curious!

Select Answer:

Visualized Solution

Visualizing the Setup

Path of the Electron

Condition to Not Hit the Wall

Calculating Maximum Speed

Substituting Magnetic Field of Solenoid

The Sigma Insight: Motion of a Charge in Magnetic Fields

Visualizing the Arena

The Dance of the Electron

The Geometric Trap

The Limiting Condition

Bringing in the Physics

The Final Piece of the Puzzle

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