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A dipole of moment vecp is placed in a u...

A dipole of moment `vecp` is placed in a uniform electric field `vecE`. The force on the dipole is `vecF` and the torque is `vec(tau)`

A

`vectau=vecp.vecE`

B

`vectau=vecpxxvecE`

C

`vectau=vecp+vecE`

D

`vectau=vecp-vecE`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of a dipole moment \(\vec{p}\) placed in a uniform electric field \(\vec{E}\), we need to analyze both the force acting on the dipole and the torque it experiences. ### Step-by-Step Solution: 1. **Understanding the Dipole**: A dipole consists of two equal and opposite charges, \(+Q\) and \(-Q\), separated by a distance \(d\). The dipole moment \(\vec{p}\) is defined as: \[ \vec{p} = Q \cdot d \] where \(d\) is the distance between the charges. 2. **Force on the Dipole**: When the dipole is placed in a uniform electric field \(\vec{E}\), the positive charge \(+Q\) experiences a force \(\vec{F}_+ = Q\vec{E}\) in the direction of the field, while the negative charge \(-Q\) experiences a force \(\vec{F}_- = -Q\vec{E}\) in the opposite direction. Therefore, the net force \(\vec{F}\) on the dipole is: \[ \vec{F} = \vec{F}_+ + \vec{F}_- = Q\vec{E} - Q\vec{E} = 0 \] Thus, the net force on the dipole in a uniform electric field is zero. 3. **Torque on the Dipole**: The torque \(\vec{\tau}\) acting on the dipole in the electric field can be calculated by considering the forces acting on the charges and their distances from the center of the dipole. The torque is given by: \[ \vec{\tau} = \vec{r} \times \vec{F} \] where \(\vec{r}\) is the position vector from the center of the dipole to the point where the force is applied. For the dipole, if it makes an angle \(\theta\) with the electric field, the torque can be expressed as: \[ \tau = pE \sin \theta \] where \(p\) is the dipole moment and \(E\) is the magnitude of the electric field. 4. **Final Expressions**: Therefore, we conclude that: - The net force on the dipole in a uniform electric field is: \[ \vec{F} = 0 \] - The torque on the dipole is: \[ \vec{\tau} = \vec{p} \times \vec{E} \] ### Summary: - The force on the dipole is zero. - The torque on the dipole is given by \(\vec{\tau} = \vec{p} \times \vec{E}\).
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