A spherical capacitor (see Ex. 26.2) has a spherical inner plate with radius a a

Question

A spherical capacitor (see Ex. 26.2) has a spherical inner plate with radius a and outer plate with radius b. The charge on the inner plate is +Q and on the outer plate it is -Q. Unlike the example, we have filled a cone shaped region of angle (0 ) with a dielectric with constant . The dielectric fills the entire volume between the two spheres inside the cone. In this problem you may neglect any fringing effects between the dielectric and the vacuum (dielectric constant = 1) that is in the rest of the capacitor. In the textbook homework this week you did problem 26.78, which said that if you put a dielectric part way into the space between two plates it can be considered as two capacitors in parallel. We’ll use that trick for this problem.

a) To begin, let’s have = 0 so that we’re just working with the standard spherical capacitor. Using Gauss’ Law, find the electric field E, potential difference V, and capacitance C for this case. The answer is in Ex. 26.2, so you must show your work to get any credit. (4 pts)

b) Show that the area of the darkly shaded region of the outer sphere is A = 2b 2 (1-cos). You can refer to problem 24.52 that you did a couple weeks ago if you need help with the geometry. (2 pts)

c) Now we have what we need to attack this problem. Ignoring any fringing, calculate the capacitance of the cone shaped region. (Hint: you’ll want to think of this as a fraction of a spherical capacitor and use the ratio of the areas.) (3 pts)

d) Repeat what you did in

c) for the region outside of the cone (i.e. the rest of the sphere), using the same trick. (3 pts)

e) Since we’re considering this as two capacitors in parallel, find the total capacitance of this system in terms of , a, b, and fundamental constants. (3 pts)

f) Find the total energy stored in the capacitor, U, when it has charge Q on it as a function of . (3 pts)

g) For what value of is the energy stored at a maximum? For what value is it at a minimum? What does this tell us about how dielectrics can be useful in electronic devices if we want to minimize energy consumption? (3 pts)

PLEASE ANSWER FULL QUESTION Including parts f and g

Explanation / Answer

Given spherical capacitor

inner plate radius = a

outer plate radius = b

CHarge = Q

dielectric constant = k'

half angle of the cone shape = theta

a. for theta = 0, we have normal spherical capacitor

using gauss law for a spherical gaussean surface at radius r, r < b, r > a

electric field inside this region = E

E*4*pi*r^2 = Q/epsilon

E = kQ/r^2 [ where k is coloumbs constant]

now, E = -dV/dr

so, dV = -kQ*dr/r^2

so integrating from r = b to r = a

V = kQ(1/a - 1/b) = kQ(b - a)/ab

now for a capacitor with capacitance C

Q = CV

hence C = Q/V = Qab/kQ(b - a) = ab/k(b - a) [ where k is coloumbs constant]

b. if we consider a ring on the outer surface at half angle theta

thickness of ring, dx = b*d(theta)

area of this ring = 2*pi*b^2*sin(theta)*d(theta)

total area of the shaded region, integrate dA form theta = 0 to theta = theta

so, A = 2*pi*b^2*(cos(0) - cos(theta)) = 2*pi*b^2*(1 - cos(theta))

c. capacitance of this shaded area be C'

then C'/C = A/A'

where A' is area of the outer sphere = 4*pi*b^2

so, C' = 2*pi*b^2*(1 - cos(theta)) *C/4*pi b^2 = (1 - cos(theta))*ab/2k(b - a)

d. for region outside the cone, let capacitance be C"

C"/C = (4pi*b^2 - 2*pi*b^2*(1 - cos(theta)))/4*pi*b^2 = (1 + cos(theta)))/2

C" = (1 + cos(theta))*ab/2k(b - a)

e. as these two capacitros are in parallel, and the smaller one has a dielectric of dielectric constant k'

net capacitance = k'*(1 - cos(theta))*ab/2k(b - a) + (1 + cos(theta))*ab/2k(b - a)

f. energy stored in a capacitor is given by E = 0.5CV^2 = 0.5Q^2/C

so for stored charge Q

U = 0.5Q^2/[k'*(1 - cos(theta))*ab/2k(b - a) + (1 + cos(theta))*ab/2k(b - a)]

U = 0.5Q^2*2k(b - a)/ab[k'*(1 - cos(theta)) + (1 + cos(theta))]

here k is coloumbs constant and k' is the dielectric constant

g. U = 0.5Q^2*2k(b - a)/ab[k'*(1 - cos(theta)) + (1 + cos(theta))]

this U is maximum when denominator is minimum, U is minimum when denominator is maximum

so, consider the denominator

D = ab[k'*(1 - cos(theta)) + (1 + cos(theta))]

applygin condition for maxima minima

d(D)/d(theta) = 0 for maxima and minima

d(ab[k'*(1 - cos(theta)) + (1 + cos(theta))])/d(theta) = 0

[k'*( sin(theta)) + ( -sin(theta))] = 0

k'*sin(theta) = sin(theta)

sin(theta)[k - 1] = 0

so theta = 0 or 180 deg

consider D = (ab[k'*(1 - cos(theta)) + (1 + cos(theta))])

for theta = 0

D = 2ab

for theta = 180

D = (ab[k'*(1 - cos(theta)) + (1 + cos(theta))]) = 2ab*k'

as k' > 1

so the energy in capacitor is maximum for theta = 0, or 360 deg

energy is minium for theta = 180 deg

hence dielectrics can be used in capacitors to make variable capacitors that can regulate the energy they store which can then be used for different purposes   

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