ctrostatics, Gauss’s Law, and Potential

**Shay10825** · Jan16-06, 12:42 AM

Hello. I need some help with the following problems.
4. A solid spherical container has a radius of 15 cm. The electric field 30 cm from the center of this sphere has a magnitude of 800 N/C. What is the surface charge density (sigma) on the sphere?
I did:
EA= q/ e0
E(4)(pi)(r^2) = [sigma(4)(pi)(r^2)] / e0
E= sigma / e0
800(8.85 X 10^-12) = sigma
sigma = 7.08 X 10^-9
But the answer is 2.8 X 10^ -8 C/m^2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
6. A uniform line charge of linear charge density lambda= 5 nC/m extends from x=0 to x=10 m. The magnitude of the electric field at the point y=12 m on the perpendicular bisector of the finite line of charge is?
I did:
I made a right triangle where 10 m was the length of the horizontal side and 12 m was the length of the vertical side. Then I found the hypotenuse to be 15.62 m
E= lambda / [(2pi)(e0)(r)]
E= (5X 10 ^ -9) / [(2pi)(8.85 X10^-112)(15.62)
E=7.49
But the answer is 2.88 N/C
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
16. An infinitely long cylindrical shell of radius 6.0 cm carries a uniform surface charge density sigma = 12 nC/m^2. The electric field at r = 6.1 cm is approximately?
I did:
EA = q/e0
E(2)(pi)(r)(l) = [sigma(2)(pi)(r)(l) ] / e0
E= sigma / e0
E= 1355.93
But the answer is 1.3 kN/c
I don’t know if my answer is wrong or correct because of rounding.
Any help would be greatly appreciated.
Thanks

**Psi-String** · Jan16-06, 01:20 AM

For first question
$LaTeX Code: \\oint \\vec{E}\\cdot d\\vec{A}=\\frac{q}{\\epsilon}$
so $LaTeX Code: q=\\epsilon EA$ where $LaTeX Code: {A}=4 \\pi {R}^2$ R= 0.3 m
thus
$LaTeX Code: \\sigma =\\frac {q}{4\\pi {r}^2}$
where r= 0.15 m

**Psi-String** · Jan16-06, 02:56 AM

For second question
$LaTeX Code: E= \\frac{2}{4\\pi \\epsilon_0} \\int_0^5 \\frac{\\lambda dx}{x^2+y^2}\\frac{y}{\\sqrt{x^2+y^2}}$

both the answer of q1 and q2 are correct

**Shay10825** · Jan16-06, 04:50 PM

Originally Posted by Psi-String

For second question
$LaTeX Code: E= \\frac{2}{4\\pi \\epsilon_0} \\int_0^5 \\frac{\\lambda dx}{x^2+R^2}\\frac{R}{\\sqrt{x^2+R^2}}$
both the answer of q1 and q2 are correct

How did you get that equation? And what do I plug in for x and R? And what do you mean by "both the answer of q1 and q2 are correct"?

**Psi-String** · Jan16-06, 10:50 PM

I mean that the answer of q1 is $LaTeX Code: 2.8 \\times 10^ -8 C/m^2$ and the anwer of q2 is

The R in the equetion represented for y which is 12m. I get this equetion by $LaTeX Code: \\int \\frac{1}{4 \\pi \\epsilon_0} \\frac{dq}{r^2}$ where r is the distance between dq and

and thus

.
And because of symmetery, we can notice that the horizontal component will be cancled so we just have to integrate the vertical component. That's why $LaTeX Code: \\frac{y}{\\sqrt{x^2+y^2}}$ appeared.

**Gamma** · Jan17-06, 01:36 AM

For the last one, You seem to use the same radius. Electric filed should be

E = sigma r / e.R where r= 6cm, and R= 6.1 cm. e=permitivity of free space.

You will get your answer in N/C. To change to kN/C devide your answer by 1000.

E = 1334 N/C or 1.3 kN/C.

Note that you made the same mistake in your first problem where

E should have been = r^2 sigma / R^2 e0

Learn how to use the gauss' low again because it is important to know the basics.