Parallel plate capacitor and dielectric

[scholio]

Homework Statement

a parallel-plate capacitor (2 plates) is filled with titanium dioxide which has a dielectric constant of 80. the area of each plate is 1 cm^2.

a) what must be the spacing between the plates if one wishes to make a 10^-10 farad capacitor?

b) if the dielectric strength of titanium dioxide is 10^8, what is the maximum chrage that this capacitor can safely store?

Homework Equations

dielectric constant kappa = 80
epsilon_0 -- constant = 8.85*10^-12

capacitance C = kappa[(epsilon_0*A)/d] where A is area, given = 1 cm sq, d is distance, unknown.

The Attempt at a Solution

part a:

capacitance C = kappa[(epsilon_0*A)/d] where C = 10^-10 farad, kappa = 80, epsilon_0 = 8.85*10^-12 ---> solve for d, d = 0.0708 meters = 7.08 cm

correct?

part b:

unsure, how does dielectric strength relate to dielectric constant?
what equation should i use, is it similar to one the i used in part a?

thanks

 

[rl.bhat]

Dielectric strength is the maximum electric field that an be applied across the plates before the dielectric breaks down.

 

[alphysicist]

Scholio,

Part a does not look correct to me. (Did you convert the area correctly to m^2? It is not 0.01 m^2.)

 

[scholio]

alphysicist, isn't 0.01m = 1cm? do i need to square it?

also i tried part b, assuming dielectric strength(= 10^8 V/m) = max electric field, i used this eq:

electric field E = kq/r^2 where k is constant 9*10^9, r is distance -- from part a =0.0708m, and then solved for q

so q = E*r^2/k = [(10^8)(0.0708^2)]/(9*10^9) = 5.57*10^-5 coulombs

correct?

 

[alphysicist]

alphysicist, isn't 0.01m = 1cm? do i need to square it?

That's right. 1 cm = 0.01m, but what you have here is 1 cm^2, which you might think of as 1 cm cm. A couple of different ways to think the conversion:

$$1 cm^2 \left(\frac{1 m}{100 cm}\right)^2 = 1\times 10^{-4} m^2$$

or equivalently:

$$1 cm\cdot cm \left(\frac{1 m}{100 cm}\right) \left(\frac{1 m}{100 cm}\right) = 1\times 10^{-4} m^2$$

also i tried part b, assuming dielectric strength(= 10^8 V/m) = max electric field, i used this eq:

electric field E = kq/r^2 where k is constant 9*10^9, r is distance -- from part a =0.0708m, and then solved for q

so q = E*r^2/k = [(10^8)(0.0708^2)]/(9*10^9) = 5.57*10^-5 coulombs

correct?

A capacitor has its charge spread out over a surface. The charge is not a point charge, and so you cannot use Coulomb's law kq/r^2. For a parallel plate capacitor, we say that the electric field is (approximately) constant. What's the relationship between potential and electric field when the field is constant?

 

[scholio]

i changed my area from 0.01 to 1*10^-4m^2 and got a distance, for part a, to be 7.08*10^-4 meters

part b:
as for be, i want to use the eq: capacitance C = epsilon_0*A/d where epsilon_0 = 8.85*10^-12, A = 1*10^-4 m^2, d = 7.08*10^-4m

but the equation is pretty useless if i know all the unknowns. i need an equation with electric field E = dielectric strength because that is given as 10^8 V/m

is potential = electric field when the field is constant? I'm not too sure, couldn't find an equation relating the two.

 

[alphysicist]

I noticed that you had the relationship between field and potential in another thread:

$$\Delta V = - \int\limits_a^b \vec E \cdot d\vec r$$

What does the right side simplify to when E is a constant field?

Once you have that, how is V and Q related for a capacitor?

 

[scholio]

if i did the integral holding E constant would i get r_b - r_a for the right side?

as for the relationship with V and q for a capacitor, capaciance C = Q/V so if i sub in r_b - r_a for V in the C eq, i'd get C = Q/ (r_b - r_a)

correct?

 

[alphysicist]

if i did the integral holding E constant would i get r_b - r_a for the right side?

To be more precise, the E would come out of the integral, and r_b-r_a is the length being integrated over, so the magnitude of the potential would be:

$$\Delta V = E d$$

(The general formula for a constant electric field is:

$$\Delta V = - E d \cos(\theta)$$

where $\theta$ is the angle between the field and displacement.)

So you know here that V= Ed, which as you say is used in C=Q/V.

as for the relationship with V and q for a capacitor, capaciance C = Q/V so if i sub in r_b - r_a for V in the C eq, i'd get C = Q/ (r_b - r_a)

correct?

 

[scholio]

so since V = Ed, and C = Q/V ---> C = Q/Ed where C = 10^-10 farad, E = 10^8 V/m, d = 7.08*10^-4 meters

so Q = 7.08*10^-6 coulombs, seems a little small

correct?

 

[alphysicist]

That looks right to me.