Problem 65 - Master Resource Book in JEE Main Physics

Question

An electric line of force in the \(x y\)-plane is given by equation \(x^{2}+y^{2}=1 .\) A particle with unit positive charge, initially at rest at the point \(x=0, y=0\) in the \(x y\)-plane (a) not move at all (b) will move along straight line (c) will move along the circular line of force (d) information is insufficient to draw any conclusion

Step-by-Step Solution

Answer

(a) not move at all

Step 1: Understand the Electric Line of Force

The equation given, \(x^{2} + y^{2} = 1\), represents a circle with a radius of 1 centered at the origin in the \(xy\)-plane. This circle is the path along which the electric line of force is said to be directed.

Step 2: Analyze the Initial Particle Position

The particle is initially at the point \((0, 0)\), which is the center of the circle (line of force) described by the equation. At this position, the particle is at the origin where no electric field is present in this setup.

Step 3: Determine the Movement of the Particle

Given the initial position \((0,0)\), and knowing that the equation represents a line of force, a particle at the center experiences no net electric field to induce movement. Line of forces suggest how a charged object would move if placed directly on the line, but since the particle is at the center, it feels no force since the electric field, assuming a radial symmetry, is zero at this point.

Step 4: Conclude on Particle's Motion

Since the particle is at rest at the origin and there is no force acting on it (as it is not on the line defined by \(x^2+y^2=1\)), it will not move from its initial position.

Key Concepts

Electric Line of Force

The concept of "electric line of force" is pivotal in understanding electric fields. Imagine these lines as representing the direction an electric field takes. In simple terms, they show how a charged particle would naturally move in that field. In our example, the equation \(x^{2} + y^{2} = 1\) describes a circle in the \(xy\)-plane. Each point on this circle can be thought of as a location where the electric field would push or pull a charge.

Electric lines of force exist in such a way that electric field vectors are tangent to these lines at any point. This means, theoretically, if a positive charge were placed anywhere on the circle, it would feel a force tangential to the circle's path. However, at the very center of the circle, like the point (0,0) in our exercise, these lines converge or balance out, creating a zone with no net force.

• Electric lines of force never intersect, which helps in visualizing the unique force direction at each point in a field.
• Lines are denser where the field is stronger, implying more force per unit charge.

Particle Motion in Electric Field

When a particle is placed in an electric field, its motion is dictated by the forces exerted by the electric field lines. A positive charge moves along the lines of force, while a negative charge moves in the opposite direction. In our case, we have a particle with a unit positive charge at the origin (0,0).

Because this point is the center of the circle defined by the electric line of force \(x^{2} + y^{2} = 1\), the particle does not lie on the electric line itself. It feels no substantial effect from the electric field, as these lines indicate the field's direction but also highlight areas of zero force when not directly on the path. Given it starts at rest and at a position where the field doesn't act, it remains motionless.

• For motion to occur, a net force must be present.
• A particle on the line of force would potentially experience motion if started off at a position with a detectable force direction.

Circular Electric Field

A circular electric field, such as the one described by \(x^{2} + y^{2} = 1\), is a simple yet powerful concept. It demonstrates how electric field lines can form closed loops. This circle suggests that if a charge were placed on it, it would experience a force such that it would tend to move around the path of the circle.

With such fields, the symmetry results in a unique condition at the center (0,0). Here, any potential forces from the surrounding electromagnetic environment balance out perfectly, leaving any charged particle within this center unaffected by the circle's field. This type of field setup, while theoretically straightforward, serves as a ground for more complex electromagnetism studies.

• It provides insights into the symmetry of force distributions.
• Such setups are analogs for many natural and artificial electromagnetic systems.