An electric dipole with dipole moment 5 × 10^-6 Cm is aligned with the direction of a uniform electric field of magnitude 4 × 10⁵N/C. The dipole is then rotated through an angle of 60° with respect to the electric field. The change in the potential energy of the dipole is:

For an electric dipole placed in a uniform electric field, the potential energy $U$ is

$U\; =\; -pE\backslash cos\backslash theta$

where

$p$ is the magnitude of the dipole moment, $E$ is the field magnitude, and $\backslash theta$ is the angle between the dipole axis and the field direction.

Initially the dipole is aligned with the field, so $\backslash theta\_i\; =\; 0^\backslash circ$. The initial potential energy is

$U\_i\; =\; -pE\backslash cos0^\backslash circ\; =\; -pE$

After rotating the dipole by $60^\backslash circ$, the new angle is $\backslash theta\_f\; =\; 60^\backslash circ$. The final potential energy becomes

$U\_f\; =\; -pE\backslash cos60^\backslash circ\; =\; -pE\backslash left(\backslash frac\{1\}\{2\}\backslash right)\; =\; -\backslash frac\{1\}\{2\}pE$

The change in potential energy is the difference

$\backslash Delta\; U\; =\; U\_f\; -\; U\_i\; =\; \backslash left(-\backslash frac\{1\}\{2\}pE\backslash right)\; -\; \backslash left(-pE\backslash right)\; =\; \backslash frac\{1\}\{2\}pE$

Now insert the given numerical values:

$p\; =\; 5\; \backslash times\; 10^\{-6\}\backslash \; \backslash text\{C·m\}$

$E\; =\; 4\; \backslash times\; 10^\{5\}\backslash \; \backslash text\{N/C\}$

Calculate the product $pE$:

$pE\; =\; \backslash left(5\; \backslash times\; 10^\{-6\}\backslash right)\backslash !\backslash left(4\; \backslash times\; 10^\{5\}\backslash right)\; =\; 20\; \backslash times\; 10^\{-1\}\; =\; 2.0\backslash \; \backslash text\{J\}$

Finally, half of this product gives the change in energy:

$\backslash Delta\; U\; =\; \backslash frac\{1\}\{2\}\backslash times\; 2.0\backslash \; \backslash text\{J\}\; =\; 1.0\backslash \; \backslash text\{J\}$

Therefore, the dipole’s potential energy increases by 1.0 J when it is rotated through 60°.