A 40.0 -cm-diameter circular loop is rotated in a uniform...

A 40.0 -cm-diameter circular loop is rotated in a uniform electric field until the position of maximum electric flux is found. The flux in this position is measured to be $5.20 \times$ $10^{5} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C} .$ What is the magnitude of the electric field?

Key Concepts

Uniform Electric Field

A uniform electric field is characterized by having the same magnitude and direction at every point in space. This uniformity simplifies calculations of electric flux because the electric field does not vary over the surface, allowing the flux to be directly computed as the product of the electric field magnitude and the area component perpendicular to the field.

Maximum Flux Condition

The maximum flux condition occurs when the orientation of the surface is such that its area vector is parallel to the electric field. In this alignment, the cosine of the angle between them is one, and the flux is maximized, being simply the product of the electric field magnitude and the area. This concept is fundamental in using flux measurements to determine the field strength.

Electric Flux

Electric flux is a scalar quantity that represents the number of electric field lines passing through a given surface. It is calculated as the dot product of the electric field vector and the area vector of the surface, and it depends on the magnitude of the field, the size of the area, and the cosine of the angle between the field and the normal to the surface.

Area of a Circle

For a circular loop, the area is an important factor in determining the total flux through the loop. The area is calculated using the formula A = ?r², where r is the radius of the circle. This area is crucial when relating the measured flux to the magnitude of a uniform electric field.

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