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A uniform spherically symmetric surface charge is initially at rest, distributed on a sphere of radius R0. Due to mutual electrostatic repulsion the shell begins to expand. Which graph best represents the speed V of the expanding shell as a function of its instantaneous radius R(t)? (The shell starts at rest at R0, accelerates, and approaches a maximum terminal speed as R -> infinity.)

1. V starts from a nonzero value at R0 and decreases to zero as R increases
2. V starts at zero at R0, increases, and saturates to a constant maximum as R -> infinity
3. V increases linearly without bound as R increases
4. V starts at zero and increases without bound (no saturation) as R increases

Correct answer: V starts at zero at R0, increases, and saturates to a constant maximum as R -> infinity

Solution

The self-energy of a charged spherical shell is U = Q²/(8*pi*epsilon0*R). As the shell expands, U decreases and is converted to kinetic energy: (1/2)mV² = U(R0) - U(R) = (Q²/(8*pi*epsilon0))(1/R0 - 1/R). At R = R0, V = 0; as R increases, V increases; as R -> infinity, 1/R -> 0 so V approaches a finite maximum V_max = sqrt((Q²/(4*pi*epsilon0 m))(1/R0)). Thus V rises from zero and saturates to a constant value, matching the curve that starts at zero and levels off.

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