Question

A point particle of charge \( Q \) is located at \( P \) along the axis of an electric dipole 1 at a distance \( r \) as shown in the figure. The point \( P \) is also on the equatorial plane of a second electric dipole 2 at a distance \( r \). The dipoles are made of opposite charge \( q \) separated by a distance \( 2a \). For the charge particle at \( P \) not to experience any net force, which of the following correctly describes the situation?

 

• A. \( \frac{a}{r} - 20 \)
• B. \( \frac{a}{r} \sim 10 \)
• C. \( \frac{a}{r} \sim 0.5 \)
• D. \( \frac{a}{r}\ \approx 3 \)

Hint:

When solving such problems, use the symmetry of the dipoles and the condition for no net force to cancel out the electric field effects.

Answer 1

The Correct Option is D

To determine if the electric field can be zero between a point charge and a dipole system, we analyze the field contributions:

1. Setting Up the Equation:
We equate the electric field from the point charge to the combined field from the dipole:

\[ \frac{kq}{(r - a)^2} = \frac{kq}{(r + a)^2} + \frac{2kqa}{(r^2 + a^2)^{3/2}} \] where we've substituted \(\cos \theta = \frac{a}{\sqrt{r^2 + a^2}}\).

2. Simplifying the Equation:
Rearranging terms gives:

\[ \frac{1}{(r - a)^2} - \frac{1}{(r + a)^2} = \frac{2a}{(r^2 + a^2)^{3/2}} \]
Which simplifies to:

\[ \frac{4ra}{(r^2 - a^2)^2} = \frac{2a}{(r^2 + a^2)^{3/2}} \]

3. Further Reduction:
Canceling common terms and squaring both sides yields:

\[ \frac{4r^2}{(r^2 - a^2)^4} = \frac{1}{(r^2 + a^2)^3} \]
Leading to:

\[ 4r^2(r^2 + a^2)^3 = (r^2 - a^2)^4 \]

4. Dimensionless Form:
Expressed in terms of the ratio \(x = a/r\):

\[ 4(1 + x^2)^3 = (1 - x^2)^4 \]

5. Physical Interpretation:
The equation suggests \(a > r\) would be required for a solution, but this contradicts the physical setup where the point charge lies between the dipole charges. Numerical analysis gives \(x \approx 3\) as a solution, though physically unrealistic in this configuration.

Conclusion:
The electric field cannot be zero at the specified location under normal physical conditions. The mathematical solution \(a \approx 3r\) exists but doesn't correspond to a physically realizable configuration in this setup.

Final Answer:
The electric field cannot be zero in this configuration No solution exists.