An electret is similar to a magnet, but rather than being permanently magnetized, it has a permanent electric dipole moment. Suppose a small electret with electric dipole moment 1.0×10⁻⁷ C m is 25 cm from a small ball charged to +25 nC, with the ball on the axis of the electric dipole. What is the magnitude of the electric force on the ball?

Verified step by step guidance

1. Step 1: Understand the problem. The electret has a permanent electric dipole moment, and the charged ball is placed on the axis of the dipole. The goal is to calculate the magnitude of the electric force on the ball due to the dipole's electric field.
2. Step 2: Recall the formula for the electric field due to a dipole along its axis. The electric field at a distance \( r \) from the dipole along its axis is given by \( E = \frac{1}{4 \pi \epsilon_0} \cdot \frac{2p}{r^3} \), where \( p \) is the dipole moment and \( \epsilon_0 \) is the permittivity of free space.
3. Step 3: Substitute the given values into the formula for the electric field. The dipole moment \( p = 1.0 \times 10^{-7} \; \text{C} \cdot \text{m} \), the distance \( r = 25 \; \text{cm} = 0.25 \; \text{m} \), and \( \epsilon_0 = 8.85 \times 10^{-12} \; \text{C}^2 / \text{N} \cdot \text{m}^2 \). Plug these values into the formula for \( E \).
4. Step 4: Calculate the force on the charged ball using \( F = qE \), where \( q \) is the charge of the ball and \( E \) is the electric field. The charge of the ball is \( q = +25 \; \text{nC} = 25 \times 10^{-9} \; \text{C} \). Substitute \( q \) and \( E \) into the formula for \( F \).
5. Step 5: Perform the calculations step by step to find the magnitude of the force. Ensure units are consistent throughout the calculation (e.g., meters for distance, coulombs for charge). The result will give the magnitude of the electric force on the ball.

Key Concepts

Electric Dipole Moment

The electric dipole moment is a vector quantity that represents the separation of positive and negative charges in a system. It is defined as the product of the charge and the distance between the charges. In this case, the electret has a permanent dipole moment, which influences the electric field it generates and interacts with other charges, such as the charged ball.

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the magnitude of the electric force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is fundamental for calculating the force exerted on the charged ball by the electric field created by the electret.

Electric Field of a Dipole

The electric field generated by an electric dipole decreases with distance and is characterized by its dependence on the dipole moment and the distance from the dipole. For a dipole aligned with the axis, the electric field at a point along the axis can be calculated using specific formulas that incorporate the dipole moment and the distance from the dipole, which is essential for determining the force on the charged ball.