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Chapter 16: Problem 29

What are the magnitude and direction of the electric field midway between two point charges, \(-15 \mu \mathrm{C}\) and \(+12 \mu \mathrm{C},\) that are $8.0 \mathrm{cm}$ apart?

Short Answer

Expert verified
Answer: To find the magnitude and direction of the electric field midway between the point charges, follow these steps: 1. Determine the position of the midpoint: The midpoint is located at 4 cm from each charge. 2. Calculate the electric field due to each charge using the electric field formula, and find the values for E1 and E2. 3. Determine the net electric field at the midpoint: E_net = |E1 - E2|. 4. Determine the direction of the net electric field, which points towards the -15 μC charge, as it has a larger magnitude. Using the given values and formula, you can calculate the magnitude of the electric field and find that the direction is towards the -15 μC charge.

Step by step solution

01

Determine the Position of the Midpoint

To find the position of the midpoint, we'll take the average of the distances of the two charges. Since the charges are 8 cm apart, the midpoint is located at 4 cm from each charge.
02

Calculate the Electric Field due to Each Charge

Use the formula for the electric field of a point charge: \(E = k\frac{|q|}{r^2}\), where \(k = 8.9875 \times 10^9 \mathrm{N\cdot m^2 \cdot C^{-2}}\) is the electrostatic constant, q is the charge, and r is the distance between the charge and the midpoint. For the \(-15 \mu C\) charge: \(E_1 = k\frac{|-15 \times 10^{-6} C|}{(0.04 m)^2}\) For the \(+12 \mu C\) charge: \(E_2 = k\frac{12 \times 10^{-6} C}{(0.04 m)^2}\) Calculate the values for \(E_1\) and \(E_2\).
03

Determine the Net Electric Field at the Midpoint

Since the electric fields of the two charges have opposite directions, the net electric field is the difference between their magnitudes: \(E_{net} = |E_1 - E_2|\) Calculate the value for \(E_{net}\).
04

Determine the Direction of the Net Electric Field

The direction of the net electric field depends on which charge has a larger magnitude. In this case, the \(-15 \mu C\) charge has a larger magnitude, so the net electric field will point towards this charge. The magnitude and direction of the electric field midway between the two point charges are found by following these steps.

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