(II) A spherical rubber balloon carries a total charge Q...

(II) A spherical rubber balloon carries a total charge $Q$ uniformly distributed on its surface. At $t=0$ the nonconducting balloon has radius $r_{0}$ and the balloon is then slowly blown up so that $r$ increases linearly to 2$r_{0}$ in a time $T$ . Determine the electric field as a function of time $(a)$ just outside the balloon surface and $(b)$ at $r=3.2 r_{0} .$

(II) A spherical rubber balloon carries a total charge $Q$ uniformly distributed on its surface. At $t=0$ the nonconducting balloon has radius $r_{0}$ and the balloon is then slowly blown up so that $r$ increases linearly to 2$r_{0}$ in a time $T$ . Determine the electric field as a function of time $(a)$ just
outside the balloon surface and $(b)$ at $r=3.2 r_{0} .$

Key Concepts

Gauss's Law

Gauss's Law states that the net electric flux through a closed surface is proportional to the enclosed charge. This principle is especially useful in systems with high symmetry, such as a uniformly charged sphere, where it simplifies the calculation of the electric field both at the surface and at external points.

Spherical Symmetry

Spherical symmetry implies that the properties of the system are uniform in all directions from the center. This symmetry allows one to treat the electric field as radially directed and dependent only on the distance from the center, which simplifies the use of Gauss's Law for finding the field.

Surface Charge Density

Surface charge density is the measure of charge per unit area on a surface. In the problem of a balloon with uniformly distributed charge, as the balloon expands the surface area increases, causing the surface charge density to decrease, which in turn affects the magnitude of the electric field.

Quasi-static Approximation

The quasi-static approximation assumes that changes in the system occur slowly enough that at each instant the system can be considered as being in equilibrium. This allows the use of static formulas like Gauss's Law at each moment during the gradual expansion of the balloon.

Electric Field of a Spherical Charge Distribution

For a spherical charge distribution with uniform surface charge, the electric field outside the sphere can be calculated as if all the charge were concentrated at the center. This concept, derived from Gauss's Law, is key to determining the electric field at any point outside the surface of the sphere.

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