Question:
An electric dipole $\vec{P} = 2qa\hat{i}$ is placed in $x$-$y$ plane, centred at the origin. Let $E_1$ and $E_2$ be the magnitudes of electric field at a point distant $r\ (\gg a)$ from its centre, along its axis and on its equatorial plane, respectively. Then the value of $\left(\dfrac{E_1}{E_2}\right)$ is:
An electric dipole $\vec{P} = 2qa\hat{i}$ is placed in $x$-$y$ plane, centred at the origin. Let $E_1$ and $E_2$ be the magnitudes of electric field at a point distant $r\ (\gg a)$ from its centre, along its axis and on its equatorial plane, respectively. Then the value of $\left(\dfrac{E_1}{E_2}\right)$ is:
Show Hint
On the axis E = 2kp/r^3 and on the equatorial plane E = kp/r^3 for r much larger than a.
Updated On: Oct 21, 2026
- $\dfrac{1}{4}$
- $\dfrac{1}{2}$
- $2$
- $4$
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
The field of an electric dipole at a far point depends on the direction. We compare the field on the axis (\(E_1\)) with the field on the equatorial plane (\(E_2\)), both at distance \(r \gg a\).
Step 2: Key Formula or Approach:
The dipole moment has magnitude \(p = 2qa\). For \(r \gg a\), the standard results are:
\[ E_{axis} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}, \qquad E_{eq} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3} \]
Step 3: Take the ratio:
The dipole lies along the x-axis, so \(E_1\) is the axial field and \(E_2\) is the equatorial field.
\[ \dfrac{E_1}{E_2} = \dfrac{2p/r^3}{p/r^3} = 2 \]
Step 4: Why the other options are wrong.
The values \(\dfrac{1}{4}\) and \(\dfrac{1}{2}\) would make the axial field smaller than the equatorial field, which is wrong. The axial field is twice as large. The value 4 would come from wrongly using \(E_{eq} = \dfrac{p}{2r^3}\).
Final Answer:
The axial field is twice the equatorial field, which is option (C). \[ \boxed{\dfrac{E_1}{E_2} = 2} \]
The field of an electric dipole at a far point depends on the direction. We compare the field on the axis (\(E_1\)) with the field on the equatorial plane (\(E_2\)), both at distance \(r \gg a\).
Step 2: Key Formula or Approach:
The dipole moment has magnitude \(p = 2qa\). For \(r \gg a\), the standard results are:
\[ E_{axis} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{2p}{r^3}, \qquad E_{eq} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{p}{r^3} \]
Step 3: Take the ratio:
The dipole lies along the x-axis, so \(E_1\) is the axial field and \(E_2\) is the equatorial field.
\[ \dfrac{E_1}{E_2} = \dfrac{2p/r^3}{p/r^3} = 2 \]
Step 4: Why the other options are wrong.
The values \(\dfrac{1}{4}\) and \(\dfrac{1}{2}\) would make the axial field smaller than the equatorial field, which is wrong. The axial field is twice as large. The value 4 would come from wrongly using \(E_{eq} = \dfrac{p}{2r^3}\).
Final Answer:
The axial field is twice the equatorial field, which is option (C). \[ \boxed{\dfrac{E_1}{E_2} = 2} \]
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