The torque on a loop of wire (a magnetic dipole) in a uniform magnetic field is \(\tau=N I A B\) sin \(\theta,\) where \(\theta\) is the angle between \(\overline{\mathbf{B}}\) and a line perpendicular to the loop of wire. Suppose an electric dipole, consisting of two charges \(\pm q\) a fixed distance \(d\) apart is in a uniform electric field \(\overrightarrow{\mathbf{E}}\). (a) Show that the net electric force on the dipole is zero. (b) Let \(\theta\) be the angle between \(\overrightarrow{\mathbf{E}}\) and a line running from the negative to the positive charge. Show that the torque on the electric dipole is \(\tau=q d E\) sin \(\theta\) for all angles $-180^{\circ} \leq \theta \leq 180^{\circ} .$ (Thus, for both electric and magnetic dipoles, the torque is the product of the dipole moment times the field strength times $\sin \theta .\( The quantity \)q d\( is the electric dipole moment; the quantity \)N I A$ is the magnetic dipole moment.)

First, let's recall that the force experienced by a charge \(q\) in a uniform electric field \(\overrightarrow{\mathbf{E}}\) is given by \(\overrightarrow{F} = q\overrightarrow{\mathbf{E}}\). As the electric dipole consists of two charges \(\pm q\), we can find the force acting on each charge individually: 1. The force on the positive charge: \(\overrightarrow{F_+} = q\overrightarrow{\mathbf{E}}\) 2. The force on the negative charge: \(\overrightarrow{F_-} = (-q)\overrightarrow{\mathbf{E}} = -q\overrightarrow{\mathbf{E}}\) Since \(\overrightarrow{F_+}\) and \(\overrightarrow{F_-}\) are equal in magnitude but opposite in direction, their sum will be zero. Therefore, the net force acting on the electric dipole is zero.

To find the torque on the electric dipole, we need to consider the individual torques acting on each of the charges due to the electric field, keeping in mind that the torque is related to the force and the distance between the charges. Let's consider the two torques: 1. The torque on the positive charge: \(\tau_+ = r_+ \times F_+ \sin\theta_+\) 2. The torque on the negative charge: \(\tau_- = r_- \times F_- \sin\theta_-\) As the angular distance between \(\overrightarrow{\mathbf{E}}\) and the line connecting the charges is given as \(\theta\), we have \(r_+ = r_- = \frac{d}{2}\) and \(\theta_+ = \theta_-\). Substituting the calculated forces \(\overrightarrow{F_+}\) and \(\overrightarrow{F_-}\), as well as the distances and angles mentioned above: 1. \(\tau_+ = \frac{d}{2} \times qE\sin\theta\) 2. \(\tau_- = \frac{d}{2} \times (-qE)\sin\theta = -\frac{d}{2}qE\sin\theta\) Now, we need to find the net torque acting on the electric dipole. Since \(\tau_+\) and \(\tau_-\) act in the same direction (both trying to orient the dipole along the electric field direction), their magnitudes will add up. Thus, $$\tau_{net} = \tau_+ + \tau_- = \frac{d}{2}qE\sin\theta - \frac{d}{2}qE\sin\theta = qdE\sin\theta$$ Therefore, the torque acting on the electric dipole in a uniform electric field is \(\tau = qdE\sin\theta\), which holds true for all angles \(-180^\circ \leq \theta \leq 180^\circ\).