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A point Q lies on the perpendicular bis...

A point `Q` lies on the perpendicular bisector of an electrical dipole of dipole moment `p`, If the distance of `Q` from the dipole is `r` (much larger than the size of the dipole), then electric field at `Q` is proportional to

A

`p^(-1)` and `r^(2)`

B

`p` and `r^(-2)`

C

`p^(2)` and `r^(-3)`

D

`p and `r^(-3)`

Text Solution

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The correct Answer is:
To solve the problem, we need to find the electric field at point Q, which lies on the perpendicular bisector of an electric dipole. The dipole moment is denoted by \( p \), and the distance from the dipole to point Q is \( r \), where \( r \) is much larger than the size of the dipole. ### Step-by-Step Solution: 1. **Understanding the Configuration**: - We have an electric dipole consisting of two equal and opposite charges separated by a small distance \( 2a \). - The dipole moment \( p \) is defined as \( p = q \cdot d \), where \( d \) is the separation between the charges. - Point Q lies on the perpendicular bisector of the dipole. 2. **Electric Field Due to a Dipole**: - The electric field \( E \) at a point on the perpendicular bisector (equatorial line) of a dipole can be expressed as: \[ E = \frac{p}{4 \pi \epsilon_0 (r^2 + a^2)^{3/2}} \] - Here, \( \epsilon_0 \) is the permittivity of free space, and \( r \) is the distance from the center of the dipole to point Q. 3. **Considering the Condition \( r \gg a \)**: - Since \( r \) is much larger than the size of the dipole, we can neglect \( a \) in the expression. Thus, we approximate: \[ E \approx \frac{p}{4 \pi \epsilon_0 r^3} \] 4. **Proportionality of Electric Field**: - From the simplified expression, we can see that the electric field \( E \) is proportional to the dipole moment \( p \) and inversely proportional to the cube of the distance \( r \): \[ E \propto \frac{p}{r^3} \] 5. **Final Conclusion**: - Therefore, we conclude that the electric field \( E \) at point Q is proportional to the dipole moment \( p \) and inversely proportional to \( r^3 \): \[ E \propto p \quad \text{and} \quad E \propto r^{-3} \] ### Final Answer: The electric field at point Q is proportional to \( p \) and \( r^{-3} \).
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