Exercise 23.48 A metal sphere with radius r a = 1.40cm is supported on an insula

Question

Exercise 23.48

A metal sphere with radius ra = 1.40cm is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius rb = 9.50cm . Charge +q is put on the inner sphere and charge ?q on the outer spherical shell. The magnitude of q is chosen to make the potential difference between the spheres 500V , with the inner sphere at higher potential.

Part A

Calculate q.

Exercise 23.48

A metal sphere with radius ra = 1.40cm is supported on an insulating stand at the center of a hollow, metal, spherical shell with radius rb = 9.50cm . Charge +q is put on the inner sphere and charge ?q on the outer spherical shell. The magnitude of q is chosen to make the potential difference between the spheres 500V , with the inner sphere at higher potential.

Part A

Calculate q.

Explanation / Answer

Spherical charge distributions are easy. The potential, relative to infinity, of a thin hollow sphere of charge Q and radius R, measured at a distance r from the center, is:

V = k Q / r ... when r >= R
V = k Q / R .... when r <= R

You can work this out from the E field, since E = -dV/dr along a radial path. The abrupt change in slope at r=R happens because the E field vanishes inside the sphere.

Potentials combine additively, so the potential inside the solid sphere is the sum:

Vinside = (k Q1 / R1) + (k Q2 / R2) ... for r <= R1 < R2

Outside of both spheres, where R1 < R2 <= r, the potential is:

Voutside = (k Q1 / r) + (k Q2 / r )

Since Q1 = q, Q2 = -q, Voutside becomes kq/r - kq/r = 0, and

Vinside = kq(1/R1 - 1/R2)

The potiential difference is ?V = Vinside - Voutside, so:
?V = kq(1/R1 - 1/R2) = kq (R2 - R1) / (R1*R2)
q = ?V * (R1 * R2) / [k * (R2 - R1) ]

All the values on the right are known or given, so it's calculator time.

Edit: Jim's answer in terms of capacitance is clever, and it would work if he used the right formula for a spherical capacitor. (His is approximately correct when the radii are nearly equal, but that's not the case here.) For the spherical capacitor, see:

http://hyperphysics.phy-astr.gsu.edu/hba...

...and remember that k = 1/(4???). You end up with the same formula as above, translated to:

q = 4??? ?V (1/R1 - 1/R2) .... replacing (1/k) and "unsimplifying" the (1/R1 - 1/R2) term.

I guess these days the 1/R1 - 1/R2 form is simpler, but I grew up in pre-calculator days when a multiplication, subtraction and an addition was better than two divisions and subtraction.

= 1/k *500* ( 1/1.4 - 1/9.5)

= 34.1589*500*.6 = 10401.7703

Related Questions

Navigate

• Browse (All)
• Browse E
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.