Question 23.37: A square metal plate of edge length 8.0 cm and negligible th......
A square metal plate of edge length 8.0 cm and negligible thickness has a total charge of 6.0 × 10^{-6} C. (a) Estimate the magnitude E of the electric field just off the center of the plate (at, say, a distance of 0.50 mm from the center) by assuming that the charge is spread uniformly over the two faces of the plate. (b) Estimate E at a distance of 30 m (large relative to the plate size) by assuming that the plate is a charged particle.
THINK To calculate the electric field at a point very close to the center of a large, uniformly charged conducting plate, we replace the finite plate with an infinite plate having the same charge density. Planar symmetry then allows us to apply Gauss’ law to calculate the electric field.
EXPRESS Using Gauss’ law, we find the magnitude of the field to be E=\sigma / \varepsilon_0 , where σ is the area charge density for the surface just under the point. The charge is distributed uniformly over both sides of the original plate, with half being on the side near the field point. Thus, \sigma=q / 2 A .
ANALYZE (a) With q=6.0 \times 10^{-6} C and A = (0.080 m)² , we obtain
\sigma=\frac{q}{2 A}=\frac{6.0 \times 10^{-6} \,C }{2(0.080 \,m )^2}=4.69 \times 10^{-4} \,C / m ^2 .
The magnitude of the field is
E=\frac{\sigma}{\varepsilon_0}=\frac{4.69 \times 10^{-4} \,C / m ^2}{8.85 \times 10^{-12} \,C ^2 / N \cdot m ^2}=5.3 \times 10^7 \,N / C .
The field is normal to the plate and since the charge on the plate is positive, it points away from the plate.
(b) At a point far away from the plate, the electric field is nearly that of a point particle with charge equal to the total charge on the plate. The magnitude of the field is E=q / 4 \pi \varepsilon_0 r^2=k q / r^2 , where r is the distance from the plate. Thus,
E=\frac{\left(8.99 \times 10^9 \,N \cdot m ^2 / C ^2\right)\left(6.0 \times 10^{-6} \,C \right)}{(30 \,m )^2}=60 \,N / C \text {. }
LEARN In summary, the electric field is nearly uniform \left(E=\sigma / \varepsilon_0\right) close to the plate, but resembles that of a point charge far away from the plate.