Problem 13 - Master Resource Book in JEE Main Physics

Question

Two point charges \(+q\) and \(-q\) are held fixed at \((-d, 0)\) and \((d, 0)\) respectively of a \((x, y)\) coordinate system, then (a) the electric field \(\mathrm{E}\) at all points on the \(x\)-axis has the same direction (b) \(\mathrm{E}\) at all points on the \(y\)-axis is along \(\hat{i}\) (c) work has to be done in bringing a test charge from infinity to the origin (d) the dipole moment is 2 qd directed along \(\hat{i}\)

Step-by-Step Solution

Answer

(d) the dipole moment is 2qd directed along \( \hat{i} \).

Step 1: Understand the Problem

Analyze the given positions of the point charges: a charge "+q" at (-d, 0) and a charge "-q" at (d, 0) . These charges form an electric dipole.

Step 2: Assess Electric Field on x-axis

On the x-axis, the electric field due to the positive charge at any point is directed away from the point charge position along the x-axis, and the one due to the negative charge is directed towards it. Thus, these vectors have opposite directions, cancelling each other out at every point on the axis except between the charges, where they reinforce. Therefore, the electric field does not have the same direction everywhere on the x-axis.

Step 3: Assess Electric Field on y-axis

On the y-axis, the electric field due to the two charges has no x-components, as they cancel out, leaving only the y-component. Since one is directed upward and the other downward, their y-components do not cancel along the y-axis, creating a resultant field that is along the vector line, which would imply a more y-dominated field. Hence, the field is not specifically along \( \hat{i} \).

Step 4: Evaluate Work Done

Bringing a test charge from infinity to the origin in the presence of a dipole involves work since the electric potential energy varies due to the influence of both charges with varying magnitudes and directions at different locations.

Step 5: Determine Dipole Moment

The dipole moment is calculated as the product of one charge and the distance between the charges, \( p = q imes 2d \). The dipole is directed from the negative to the positive charge, which aligns along the x-axis or \( \hat{i} \). Therefore, the dipole moment is 2qd \hat{i}.

Key Concepts

Electric Field

An electric field is a region around a charged particle where forces are exerted on other charges. When we analyze the electric field for an electric dipole, it becomes interesting. The dipole consists of two equal and opposite charges: one positive "+q" and one negative "-q".

On the x-axis, these charges create electric fields that have different directions. This depends on whether the point is closer to one charge or the other. Their fields cancel each other out when observed at far-off regions on the axis. In between the charges, they reinforce each other, leading to a noticeable field in that region.

On the y-axis, the story is different. Here, the electric fields from the dipole's charges also interact. However, their x-components cancel out, leaving only the y-components. This results in a resultant field that is predominantly vertical but also variable throughout the axis. It is not always aligned purely along a single direction like the y-axis.

Dipole Moment

The dipole moment is a measure of the separation of positive and negative charges within a system. For this specific exercise, the dipole moment can be calculated using the formula:

• Dipole Moment ( \( p \) ) = Charge ( \( q \) ) × Distance ( \( 2d \) )

It represents how much charge is separated by a certain distance and is a crucial concept in electric dipoles.

Dipole moments point from the negative charge to the positive charge. Hence, in this scenario, the dipole moment is expressed as 2qd \( \hat{i} \) .

This shows it is directed along the x-axis from -q to +q. Understanding the dipole moment is essential because it tells us about the charge distribution and its potential impacts on nearby charges or fields.

Work Done on Charge

When you move a test charge within an electric field, you do work. In the context of our electric dipole, bringing a test charge from infinity to a point like the origin involves traditional electrical work principles because the potential energy changes as the test charge moves through varying electric fields.

The work done is associated with:

• Potential energy differences at each point in the field
• Path taken by the test charge

In our case, the dipole affects the path significantly, as the test charge experiences forces due to both positive and negative charges. As we approach the origin, the electric field's direction and strength influenced by both charges will continually shift. Hence, the work done isn't trivial.

Studying this can highlight how the energy dissipates or is absorbed while maneuvering through such electric fields. It lays the foundation for deeper electrostatic studies and understanding forces at play in electromagnetic fields.