An insulating ball of radius carries a total charge spread uniformly through its volume

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Electrostatic fields and potential · Electrostatic fields and potential

Electricity, Magnetism and Electromagnetism, Ch Electric flux and Gauss's law

Problem

An insulating ball of radius

R=9.0\,\mathrm{cm}

carries a total charge

Q=+52\,\mathrm{nC}

spread uniformly through its volume. The charge is fixed and there is no other charge anywhere. Let

r

be the distance from the centre. ((a)) State the symmetry argument in two steps: what operations leave this charge distribution unchanged, and what those operations force upon the direction of

\mathbf{E}

and the variables it may depend on. Hence say why a sphere concentric with the ball is the right Gaussian surface and a cube is not. ((b)) Using Gauss's law, derive expressions for the magnitude of the field for

r\le R

and for

r\ge R

, in terms of

Q

R

r

and

\varepsilon_0

. ((c)) Evaluate the field magnitude at

r=4.5\,\mathrm{cm}

and at

r=18\,\mathrm{cm}

, and state the direction in each case. ((d)) Show that your two expressions agree at

r=R

and give that value.

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