Set 54 Problem number 7

Problem

An electrostatic field has uniform magnitude 8.4 N/C.  A circular region whose radius is 5.3 meters lies within the field.  What is the electrostatic flux through the region if the field makes an angle of 23 degrees with a perpendicular to the plane of the region?

Solution

The area of the region is `pi ( 5.3 m) ^ 2 = 88.2 m ^ 2, so the flux of a perpendicular 8.4 N/C field would be

• flux of perpendicular field:  flux = ( 8.4 /C)( 88.2 m ^ 2) = 740.8 Nm ^ 2/C.

The 23 degree angle reduces the flux, changing it by factor cos( 23 deg) = .9205 so the actual flux is

• flux at given angle:  flux = ( 740.8 N m ^ 2/C)( .9205) = 681.9 N m ^ 2/C.

Generalized Solution

The electrostatic flux of a field E (the field typically measured in Newtons / Coulomb) intercepted at a perpendicular by an area A is simply

• flux at perpendicular:  flux = E * A.

If the field makes angle `theta with the perpendicular, then flux is reduced according to the cosine of the angle:

• flux at angle `theta:   flux = (flux at perpendicular) * cos(`theta) = E * A * cos(`theta).