Question

In a uniform electric field of magnitude \(E\), the field lines cross through a rectangle of area \(A\) at an angle of \(60.0^{\circ}\) with respect to the plane of the rectangle. What is the flux through the rectangle?

Short answer

Answer: The electric flux through the rectangle is \(\frac{1}{2} \times E \times A\).

Step-by-step solution

Identify the given values and formula to be used

We are provided with the following information: - Magnitude of the uniform electric field, \(E\). - Area of the rectangle, \(A\). - Angle (relative to the plane of the rectangle), \(\theta = 60.0^{\circ}\). The formula needed to calculate the electric flux is: \(\Phi = EA \cos{\theta}\).

Convert the angle from degrees to radians

Before using the formula, we need to convert the angle \(\theta\) from degrees to radians: \(\theta_\text{radians} = \theta_\text{degrees} \times \frac{\pi}{180}\)

Calculate the angle in radians

Using the given angle of \(60.0^{\circ}\), we can now find the angle in radians: \(\theta_\text{radians} = 60.0 \times \frac{\pi}{180} = \frac{\pi}{3}\)

Apply the formula to find electric flux

Now let's plug the values into the flux formula, \(\Phi = EA \cos{\theta_\text{radians}}\): \(\Phi = E \times A \times \cos{(\frac{\pi}{3})}\)

Solve for the electric flux

Evaluate the expression for the electric flux: \(\Phi = EA (\frac{1}{2})\) \(\Phi = \frac{1}{2} \times E \times A\) The electric flux through the rectangle is \(\frac{1}{2} \times E \times A\).