a) An insulating sphere carries a charge that is spread uniformly throughout its volume. A con- ducting sphere of the same size carries the same charge, but it is spread over its surface only with no charge in the interior (as it must for a conductor!) Compare the electric fields outside the insulator to the field outside the conductor. Compare the electric fields inside the insulator to the field inside the conductor. (b) Part (a) is an example of a general rule: A charge distributed through some volume will always produce an electric field whose value is a continuous function of position as we cross out of the charge (as we did going from inside to outside of the uniformly charged insulating sphere.) A charge that is distributed across an area (a thin layer of charge with surface charge density o) will produce a field whose value is a DIScontinuous function of position, and the discontinuity is only in the component that is perpendicular to the area and has a value equal to o/ɛo. Show that this is true for the cylindrical symmetry and planar symmetry examples we examined (in other words, check that at r = R, E is continuous for an infinite uniformly charged cylinder and that it is discontinuous, with the discontinuity as claimed, for an infinite conducting charged tube and also when we cross an infinite sheet of charge.) You do not need to derive ANY new result - you may use the results we already have for these as your starting point, so this is not intended to be long or complicated! HINT for the thin tube: express the charge Q as surface density times area.
a) An insulating sphere carries a charge that is spread uniformly throughout its volume. A con- ducting sphere of the same size carries the same charge, but it is spread over its surface only with no charge in the interior (as it must for a conductor!) Compare the electric fields outside the insulator to the field outside the conductor. Compare the electric fields inside the insulator to the field inside the conductor. (b) Part (a) is an example of a general rule: A charge distributed through some volume will always produce an electric field whose value is a continuous function of position as we cross out of the charge (as we did going from inside to outside of the uniformly charged insulating sphere.) A charge that is distributed across an area (a thin layer of charge with surface charge density o) will produce a field whose value is a DIScontinuous function of position, and the discontinuity is only in the component that is perpendicular to the area and has a value equal to o/ɛo. Show that this is true for the cylindrical symmetry and planar symmetry examples we examined (in other words, check that at r = R, E is continuous for an infinite uniformly charged cylinder and that it is discontinuous, with the discontinuity as claimed, for an infinite conducting charged tube and also when we cross an infinite sheet of charge.) You do not need to derive ANY new result - you may use the results we already have for these as your starting point, so this is not intended to be long or complicated! HINT for the thin tube: express the charge Q as surface density times area.
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Transcribed Image Text:a) An insulating sphere carries a charge that is spread uniformly throughout its volume. A con-
ducting sphere of the same size carries the same charge, but it is spread over its surface only with
no charge in the interior (as it must for a conductor!) Compare the electric fields outside the insulator
to the field outside the conductor. Compare the electric fields inside the insulator to the field inside
the conductor.
(b) Part (a) is an example of a general rule: A charge distributed through some volume will always
produce an electric field whose value is a continuous function of position as we cross out of the
charge (as we did going from inside to outside of the uniformly charged insulating sphere.) A charge
that is distributed across an area (a thin layer of charge with surface charge density o) will produce
a field whose value is a DIScontinuous function of position, and the discontinuity is only in the
component that is perpendicular to the area and has a value equal to o/eo. Show that this is true for
the cylindrical symmetry and planar symmetry examples we examined (in other words, check that at
r = R, E is continuous for an infinite uniformly charged cylinder and that it is discontinuous, with the
discontinuity as claimed, for an infinite conducting charged tube and also when we cross an infinite
sheet of charge.) You do not need to derive ANY new result - you may use the results we already
have for these as your starting point, so this is not intended to be long or complicated! HINT for the
thin tube: express the charge Q as surface density times area.
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