Question

Two charges, \(+e\) and \(-e,\) are a distance of \(0.680 \mathrm{nm}\) apart in an electric field, \(E\), that has a magnitude of \(4.40 \mathrm{kN} / \mathrm{C}\) and is directed at an angle of \(45.0^{\circ}\) with respect to the dipole axis. Calculate the dipole moment and thus the torque on the dipole in the electric field.

Short answer

**Short Answer:** The dipole moment of the system with charges \(+e\) and \(-e\) separated by \(0.680 \ \mathrm{nm}\) is calculated to be \(1.088 \times 10^{-28} \ \mathrm{C\cdot m}\). When placed in an external electric field of magnitude \(4.40 \times 10^3 \ \mathrm{N/C}\) at an angle of \(45.0^{\circ}\) from the dipole axis, the torque experienced by the dipole is \(3.373 \times 10^{-25} \ \mathrm{N\cdot m}\).

Step-by-step solution

Calculate the dipole moment

To calculate the dipole moment, we can use the formula \(\vec{p} = q\vec{d}\), where \(q\) is the charge and \(\vec{d}\) is the vector joining the two charges. In this case, the charges are \(+e\) and \(-e\), and the distance between them is \(0.680 \ \mathrm{nm}\). We can convert the distance to meters and multiply it by the elementary charge (\(e = 1.6 \times 10^{-19} \ \mathrm{C}\)) to get the dipole moment: $$ \vec{p} = (1.6 \times 10^{-19} \ \mathrm{C}) (0.680 \times 10^{-9} \ \mathrm{m}) $$

Find the dipole moment

Now we can calculate the dipole moment by multiplying the charge and the distance: $$ \vec{p} = (1.6 \times 10^{-19} \ \mathrm{C}) (0.680 \times 10^{-9} \ \mathrm{m}) = 1.088 \times 10^{-28} \ \mathrm{C\cdot m} $$ The dipole moment is \(1.088 \times 10^{-28} \ \mathrm{C\cdot m}\).

Calculate the torque on the dipole

To calculate the torque \(\tau\) on the dipole in the electric field, we can use the formula \(\tau = pE \sin{\theta}\), where \(p\) is the magnitude of the dipole moment, \(E\) is the magnitude of the electric field, and \(\theta\) is the angle between the dipole axis and the electric field. In this case, \(p = 1.088 \times 10^{-28} \ \mathrm{C\cdot m}\), \(E = 4.40 \times 10^3 \ \mathrm{N/C}\), and \(\theta = 45.0^{\circ}\).

Find the torque

Now, we can calculate the torque by plugging the values into the formula: $$ \tau = (1.088 \times 10^{-28} \ \mathrm{C\cdot m})(4.40 \times 10^3 \ \mathrm{N/C}) \sin{45.0^{\circ}} $$ Converting the angle to radians and evaluating the sine function: $$ \tau = (1.088 \times 10^{-28} \ \mathrm{C\cdot m})(4.40 \times 10^3 \ \mathrm{N/C}) \sin{\frac{45.0\pi}{180}} $$ $$ \tau = (1.088 \times 10^{-28} \ \mathrm{C\cdot m})(4.40 \times 10^3 \ \mathrm{N/C}) \frac{\sqrt{2}}{2} $$ Finally, we can calculate the torque: $$ \tau = 3.373 \times 10^{-25} \ \mathrm{N\cdot m} $$ The torque on the dipole in the electric field is \(3.373 \times 10^{-25} \ \mathrm{N\cdot m}\).