A dipole with charges ±q and separation 2 a is located a distance x from a point charge +Q, orie

step 1 So we have plus Q charge here and minus Q charge here. The half distance is A. And here we have

A dipole with charges ±q and separation 2 a is located a...

A dipole with charges $\pm q$ and separation $2 a$ is located a distance $x$ from a point charge $+Q$, oriented as shown in Fig. $20.32$. Find expressions for the magnitude of (a) the net torque and (b) the net force on the dipole, both in the limit $x \gg a$. (c) What's the direction of the net force?

Key Concepts

Electric Dipole

An electric dipole is a pair of equal and opposite charges separated by a certain distance. It represents a simple system for studying non-uniform charge distributions and plays a fundamental role in understanding interactions with external electric fields.

Electric Dipole Moment

The electric dipole moment is a vector quantity defined as the product of one of the charges and the separation distance between the charges, directed from the negative charge to the positive charge. It quantifies the overall polarity of the dipole and determines the magnitude of its interaction with external electric fields.

Electric Field Due to a Point Charge

The electric field generated by a point charge is described by Coulomb's law and decreases with the square of the distance from the charge. This concept helps in understanding how the field varies in space and serves as the source that interacts with a dipole in many electrostatic problems.

Torque on a Dipole in an External Electric Field

When placed in an external electric field, an electric dipole experiences a torque that tends to align the dipole moment with the field. This torque is given by the cross product of the dipole moment and the electric field vector, and is fundamental in analyzing rotational dynamics in dipole-field interactions.

Force on a Dipole in a Non-Uniform Electric Field

A dipole in a non-uniform electric field also experiences a net force due to variations in the field's magnitude over the extent of the dipole. This force can be calculated using the gradient of the field and is essential for understanding translational motion of dipoles in spatially varying fields.

Far-Field Approximation

The far-field approximation (x >> a) assumes that the distance from the dipole to the point of observation is much greater than the separation of the charges in the dipole. This simplifies the analysis by allowing higher order terms to be neglected, leading to easier computation of torque and force magnitudes.

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