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The figure below shows two lines of charge in cross-section coming out of the page. Each line of charge has a linear charge density of ±3.30 nC/cm and they are separated by a distance d = 19.5 cm. What is the magnitude of the electric field at the dot, a point y = 6.70 cm above the midpoint between the lines?

Question

The figure below shows two lines of charge in cross-section coming out of the page. Each line of charge has a linear charge density of ±3.30 nC/cm and they are separated by a distance d = 19.5 cm. What is the magnitude of the electric field at the dot, a point y = 6.70 cm above the midpoint between the lines?

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Solution

To solve this problem, we need to use the principle of superposition and the formula for the electric field due to a line of charge.

Step 1: Convert the given quantities to SI units.

Linear charge density, λ = ±3.30 nC/cm = ±3.30 x 10^-9 C/m Distance, d = 19.5 cm = 0.195 m Distance, y = 6.70 cm = 0.067 m

Step 2: Calculate the distance from the point to each line of charge.

The point is located at the midpoint between the two lines of charge, so the distance from the point to each line of charge is d/2 = 0.195 m / 2 = 0.0975 m.

Step 3: Calculate the electric field due to each line of charge at the point.

The formula for the electric field due to a line of charge is E = λ / (2πε₀r), where ε₀ is the permittivity of free space (8.85 x 10^-12 C²/N·m²) and r is the distance from the line of charge.

For the positive line of charge, E₁ = (3.30 x 10^-9 C/m) / (2π * 8.85 x 10^-12 C²/N·m² * 0.0975 m) = 5.68 x 10^5 N/C.

For the negative line of charge, E₂ = -(3.30 x 10^-9 C/m) / (2π * 8.85 x 10^-12 C²/N·m² * 0.0975 m) = -5.68 x 10^5 N/C.

Step 4: Use the principle of superposition to find the total electric field at the point.

The electric fields due to the two lines of charge are in opposite directions, so they will subtract from each other. The total electric field at the point is E = E₁ + E₂ = 5.68 x 10^5 N/C - 5.68 x 10^5 N/C = 0 N/C.

So, the magnitude of the electric field at the point is 0 N/C.

This problem has been solved

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