Question

If $E_{a}$ and $E_{q}$ represent the electric field intensity due to a short dipole at a point on its axial line and on the equatorial line at the same distance '$r$' from the centre of the dipole, then}

• A. E_{a} = E_{q}
• B. E_{a} = \frac{1}{2} E_{q}
• C. E_{a} = \frac{1}{\sqrt{2 E_{q}
• D. E_{a} = 2 E_{q}

Hint:

For any short dipole at a fixed distance $r$, the field at an axial point is always double the field at an equatorial point.

Answer 1

The Correct Option is A

Step 1: The electric field intensity $E_{a}$ at a point on the axial line of a short dipole of moment $p$ at a distance $r$ from its center is given by the formula:\n\[ E_{a} = \frac{1}{4 \pi \epsilon_{0 \frac{2p}{r^{3 \]\n\n
Step 2: The electric field intensity $E_{q}$ at a point on the equatorial line of the same short dipole at the same distance $r$ from its center is given by the formula:\n\[ E_{q} = \frac{1}{4 \pi \epsilon_{0 \frac{p}{r^{3 \]\n\n
Step 3: By comparing the two expressions, we can take the ratio of the axial field to the equatorial field:\n\[ \frac{E_{a{E_{q = \frac{\frac{1}{4 \pi \epsilon_{0 \frac{2p}{r^{3}{\frac{1}{4 \pi \epsilon_{0 \frac{p}{r^{3} \]\n\[ \frac{E_{a{E_{q = 2 \]\n\n
Step 4: Rearranging the equation gives the relationship:\n\[ E_{a} = 2E_{q} \]