First slide

Electric field

Question

A positively charged sphere of radius r₀ carries a volume charge density $ρ$ (figure). A spherical cavity of radius $r_{0} / 2$ is then scooped out and left empty. C₁ is the center of the sphere and C₂ that of the cavity. What is the direction and magnitude of the electric field at point B?

Moderate

A

$\frac{17 ρ r_{0}}{54 ε_{0}} left$
B

$\frac{ρ r_{0}}{6 ε_{0}} left$
C

$\frac{17 ρ r_{0}}{54 ε_{0}} right$
D

$\frac{ρ r_{0}}{6 ε_{0}} right$

Solution

Electric field on surface of a uniformly charged sphere is given by
$\frac{Q}{4 π ε_{0} R^{2}} = \frac{ρ R}{3 ε_{0}}$
Electric field. at outside point is given by
$\begin{array}{l} E = \frac{Q}{4 π ε_{0} r^{2}} = \frac{ρ R^{3}}{3 ε_{0} r^{2}} \\ E_{B} = E_{whole sphere} - E_{cavity} \\ = \frac{ρ r_{0}}{3 ε_{0}} - \frac{ρ {(\frac{r_{0}}{2})}^{3}}{3 ε_{0} {(\frac{3 r_{0}}{2})}^{2}} = \frac{17 ρ r_{0}}{54 ε_{0}} \end{array}$

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