Problem 39 - Master Resource Book in JEE Main Physics

Question

If \(E_{a}\) be the electric field strength of a short dipole at a point on its axial line and \(E_{e}\) that on equatorial line at the same distance, then (a) \(E_{c}=2 E_{a}\) (b) \(E_{a}=2 E_{e}\) (c) \(E_{a}=E_{c}\) (d) None of these

Step-by-Step Solution

Answer

The correct option is (b): \(E_{a}=2 E_{e}\).

Step 1: Understand the Problem

We need to compare the electric field strengths of a short dipole on its axial line and equatorial line. Given electric field strengths are labeled as \(E_a\) for axial line and \(E_e\) for equatorial line.

Step 2: Recall Formula for Axial Line

The electric field on the axial line of a short dipole at a distance \(r\) is given by:\[ E_a = \frac{1}{4\pi \varepsilon_0} \cdot \frac{2p}{r^3} \]where \(p\) is the dipole moment and \(\varepsilon_0\) is the permittivity of free space.

Step 3: Recall Formula for Equatorial Line

The electric field on the equatorial line of a short dipole at the same distance \(r\) is:\[ E_e = \frac{1}{4\pi \varepsilon_0} \cdot \frac{p}{r^3} \]

Step 4: Compare the Two Electric Fields

Compare the expressions for \(E_a\) and \(E_e\). Note that \(E_a = 2 \cdot E_e\) as \(\frac{2p}{r^3} = 2 \times \frac{p}{r^3}\).

Step 5: Identify the Correct Option

Based on the comparison, find that \(E_a = 2 E_e\). Thus, the correct option is (b).

Key Concepts

Axial Line Electric Field

When we talk about an axial line, we refer to a line extending through the dipole and along its axis. Imagine a dipole as a pair of charges that are equal and opposite. If we place a point on this line, far away yet still on the axis, it experiences the electric field generated by the dipole.
The electric field, denoted as \(E_a\), is calculated using the formula:

• \(E_a = \frac{1}{4\pi \varepsilon_0} \cdot \frac{2p}{r^3}\)

In this formula, \(p\) stands for the dipole moment and \(r\) is the distance of the point from the dipole's center. The factor \(\frac{1}{4\pi \varepsilon_0}\) is a constant that accounts for the permittivity of free space.
This line allows the electric field to have a particular strength that depends on the position along it. Here, the triple dependency on the distance \(r\) means the electric field decreases quite rapidly as we move away from the dipole.

Equatorial Line Electric Field

The equatorial line of a dipole runs perpendicular to the axial line, passing through the midpoint between the two charges. At any point along this line, the electric field is different from that on the axial line.
In the case of the equatorial line, the electric field at a point is given by:

• \(E_e = \frac{1}{4\pi \varepsilon_0} \cdot \frac{p}{r^3}\)

Here, you can see that the formula is quite similar to that of the axial line, but it lacks the factor of 2. This means that for the same distance \(r\), the electric field on the equatorial line is half of the strength of that on the axial line. This relationship is crucial when comparing these two field strengths.

Dipole Moment

The dipole moment is a vector quantity symbolized by \(p\). It represents the strength of the dipole and its orientation in space. The dipole moment is calculated as the product of the charge \(q\) and the separation distance \(d\) between the two charges:

• \(p = q \cdot d\)

The dipole moment plays a significant role in determining the electric fields that a dipole generates. It influences the fields on both the axial and equatorial lines. The direction of the dipole moment vector usually extends from the negative charge towards the positive charge.
Understanding the dipole moment helps explain why we see different electric fields at different points around a dipole.