Electric Dipoles | Brilliant Math & Science Wiki

If an electric dipole is placed in an external electric field the charges will feel forces in opposite directions by virtue of their opposite signs, causing a rotation if the two charges remain bound to one another. Hence, the forces actually manifest themselves as a torque.

Since the torque is largest when \(\vec{p}\) is perpendicular to \(\vec{E}\) and decreases as the two vectors become more parallel, it is best modeled as the cross product between the two vectors.

The torque on a dipole \(\vec{p}\) placed in an external field \(\vec{E}\) is

\[\tau = \vec{p} \times \vec{E}\]

Two charges \(+Q\) and \(-Q\) are placed at \(x = 1\text{ m}\) and \(x = -1\text{ m}\) respectively in an external field \(\vec{E} = 3 \hat{x} + 4 \hat{y}.\) What is the torque on the dipole?

In order to evaluate the cross product, write down the dipole.

\[\vec{p} = q\vec{d} = (Q) (2 \hat{x}) = 2Q\hat{x}\]

Now evaluate the cross product.

\[\tau = \vec{p} \times \vec{E} =( 2Q\hat{x}) \times (3 \hat{x} + 4 \hat{y}) = (8Q \text{ Nm})\hat{z}\]

-AB

None of these choices

AB

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What is the torque on an ideal electric dipole with moment \(\vec{p} = -A \hat{z}\) placed in an external electric field \(\vec{E} = B \hat{z}?\)