Problem 107 - A Complete Resource Book in Physics for JEE Main

Question

Three points \(A, B\), and \(C\) are at a distance of \(1 \mathrm{~m}, 2 \mathrm{~m}\), and \(\mathrm{lm}\) from an infinitely long charged wire of linear charge density \(\lambda \mathrm{C} / \mathrm{m}\). A charge \(q\) is taken from \(A\) to \(B, B\) to \(C\), and finally \(C\) to \(A\). Which of the following is/are correct about the work done in the above process? (A) \(W_{A B}=2 W_{B C}\) (B) \(W_{A B}=-W_{B C}\) (C) \(W_{B C}=0\) (D) \(W_{C A}=0\)

Step-by-Step Solution

Answer

The correct statement(s) is/are (B) \(W_{AB} = - W_{BC}\) and (D) \(W_{CA} = 0\).

Step 1: Determine the electric field due to an infinitely long charged wire

The electric field E due to an infinitely long charged wire with linear charge density λ at a perpendicular distance 'r' from the wire is given by: \(E = \frac{2 \pi \epsilon_0}{\lambda} * r\)

Step 2: Calculate the potential difference ΔV for each path A→B, B→C, and C→A

First, we need to determine the potential difference ΔV for each of the given paths. Let's calculate the electric field at each point A, B, C: Electric field at A (E_A) = \(\frac{2 \pi \epsilon_0}{\lambda} * 1\) Electric field at B (E_B) = \(\frac{2 \pi \epsilon_0}{\lambda} * 2\) Electric field at C (E_C) = \(\frac{2 \pi \epsilon_0}{\lambda}\) (as E at infinity is considered to be zero) Now, we have the electric field at each point, and can find the potential difference: ΔV_AB = V_B - V_A ΔV_BC = V_C - V_B ΔV_CA = V_A - V_C

Step 3: Compute the work done for each path A→B, B→C, and C→A

Now, we have potential differences for each path, and we can compute the work done using the formula: Work done (W) = q * ΔV Work done A→B = \(q * ΔV_{AB}\) Work done B→C = \(q * ΔV_{BC}\) Work done C→A = \(q * ΔV_{CA}\)

Step 4: Determine if the given statements are true

Finally, let's check whether the given statements are true or false. (A) \(W_{AB} = 2 W_{BC}\) From the work done formulas in step 3, we can see that this statement is False. (B) \(W_{AB} = - W_{BC}\) From the work done formulas in step 3, we can see that this statement is True. (C) \(W_{BC} = 0\) From the work done formulas in step 3, we can see that this statement is False. (D) \(W_{CA} = 0\) From the work done formulas in step 3, we can see that this statement is True. So, the correct statement(s) is/are (B) \(W_{AB} = - W_{BC}\) and (D) \(W_{CA} = 0\).

Key Concepts

Electric Field Due to Charged Wire

Understanding the electric field generated by an infinitely long charged wire is crucial in electrostatics. The expression for the electric field at a given distance from such a wire is derived using Gauss's Law, a fundamental principle in electromagnetism. For a wire with linear charge density \(\lambda\), the electric field strength (E) at a distance (r) is given by the formula:
\[E = \frac{2\lambda}{4\pi \epsilon_0 r}\]
This equation tells us that the electric field is directly proportional to the linear charge density of the wire and inversely proportional to the distance from the wire. To visualize this, think of the electric field as being stronger closer to the wire and weakening as you move further away. For students, a useful tip is to remember that this expression yields the electric field's magnitude, not its direction, which is radially outward from the wire if \(\lambda\) is positive, and inward if negative.

When we consider the movement of a test charge in this electric field, we see that work is done on or by the charge as it moves from one point to another. Intuitively, as the distance from the wire changes, so does the electric field's strength experienced by the charge, and hence the amount of work done. For instance, moving a charge closer to a charged wire requires work against the electric field, which would be portrayed in an increase in the electric potential encountered—leading us to our next concept.

Electric Potential Difference

The electric potential difference, commonly referred to as voltage, is a measure of the work done per unit charge as a charge moves between two points within an electric field. The potential difference between two points A and B is calculated as the work done to move a unit positive charge from A to B.

\[\Delta V = V_B - V_A\]
It is key for students to grasp that when a charge moves in the direction of decreasing potential, work is done by the electric field, and energy is released. Conversely, when moving against the electric field direction, energy is consumed as work is done against the field. In the context of our infinite wire, the potential difference due to this wire can be calculated by integrating the electric field along the path between two points. This brings us to the broader principle that encompasses the relation between work and energy in electrostatics, the work-energy principle.

Work-Energy Principle in Electrostatics

The work-energy principle in electrostatics is an essential concept that explains how work done by or against an electric field results in a change in the electric potential energy of a charge. It's stated as follows:

\[W = q \Delta V\]
The equation connects the work (W) done on a charge (q) to the potential difference (\(\Delta V\)) it moves through. This principle tells us that the work done by the electric field is a measure of the change in potential energy of the charge. Remember, if the charge is moved in a closed loop within an electric field, like from point A to B to C and back to A as in our exercise, the net work done by the field is zero. This is because the charge ends up with the same potential energy it started with—no energy is lost or gained over a complete cycle.

For our given problem, it means that the work done moving from C to A (\(W_{CA}\)) is zero, as there is no net change in potential energy for the round trip. This illustrates conservation of energy in electrostatic processes and helps explain why some of the work-related statements in the exercise are true while others are not. Reiterating for learners, energy is always conserved in electrostatic interactions, which is fundamental to solving problems related to electrostatics.