Motion In A Magnetic Field

To understand this, we need to recall what the slope of a velocity-time (v–t) graph represents.
Slope of v–t graph = acceleration
Now consider the given condition:
- The velocity-time graph is a straight line inclined to the time axis.
- A straight line means the slope is constant.
So if the slope (i.e., acceleration) is constant, this implies:

Let’s analyze the options:
- (A) Uniform velocity: would mean the v–t graph is a horizontal straight line (slope = 0).
- (C) Variable acceleration: would require the v–t graph to be curved.
- (D) Zero acceleration: would again correspond to a horizontal line (zero slope).
Only option (B) correctly matches a straight-line v–t graph with non-zero slope.

Step 1: Understanding the Concept:
This problem combines two key concepts: the work-energy theorem for a charged particle in an electric field, and the dynamics of a charged particle in uniform circular motion in a magnetic field. We need to find a relationship between the radius of the circular path and the initial accelerating voltage and final speed.

Step 2: Key Formula or Approach:
1. When a particle of charge is accelerated by a potential difference , the kinetic energy gained is equal to the work done: . Also, . Therefore, . 2. When this particle enters a magnetic field perpendicularly, the magnetic force provides the centripetal force for circular motion: . 3. From this second equation, the radius of the circular path is: .

Step 3: Detailed Explanation:
We have two main equations: (i) (ii) The question asks for the proportionality of in terms of and . We need to combine the equations to find this relationship. Let's look at the expression in option (A), which is . From equation (i), we can write: Now let's compare this with our expression for the radius from equation (ii): We can see that both and are directly proportional to the term . We can write in terms of : Since the magnetic field is uniform and constant, we have the proportionality:

Step 4: Final Answer:
The radius of the circular path is proportional to .