# The electric field of two plates

#### mborn

Another question for Gauss' Law;

A very large, thin, flat plate of Aluminium of area A has a total charge Q uniformly distributed over its surfaces. If the same charge is spread uniformly over the upper surface of an otherwise identical glass plate, compare the lectric fields just above the centers of the upper surface of each plate.

M B

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#### hhegab

Hi,

For Al $$E= \frac{\sigma}{\epsilon_o}$$
for Glass $$E= \frac{\sigma}{2\epsilon_o}$$

Someone tell me if I am wrong, please

#### Andrew Mason

Homework Helper
hhegab said:
Hi,

For Al $$E= \frac{\sigma}{\epsilon_o}$$
for Glass $$E= \frac{\sigma}{2\epsilon_o}$$

Someone tell me if I am wrong, please

The charge Q spreads out over both surfaces of the aluminum and there is 0 charge inside the metal. So $\sigma = Q/2A$

The field of the Al sheet is $$(E_{top} + E_{bot}) = \frac{Q/2A}{\epsilon_0}$$

So: $$E_{top} = \frac{Q/2A}{2\epsilon_0}[/itex] For the glass plate, Q is distributed over area A (not 2A), so $\sigma = Q/A$ [tex]E_{top} = \frac{Q/A}{2\epsilon_0}[/itex] AM #### Gokul43201 Staff Emeritus Science Advisor Gold Member hhegab's right...though I wish he hadn't spilled the solution out. AM, I believe you have it backwards. In both cases, the surface charge density is usually defined as $\sigma = Q/A$ For Al : [tex]\vec{E} = \vec{E1} + \vec{E2} = \sigma /2 \epsilon _0 + \sigma /2 \epsilon _0 = \sigma /\epsilon _0$$
For glass, there's only the one charged surface.

#### Andrew Mason

Homework Helper
Gokul43201 said:
hhegab's right...though I wish he hadn't spilled the solution out.

AM, I believe you have it backwards. In both cases, the surface charge density is usually defined as $\sigma = Q/A$
For Al :
$$\vec{E} = \vec{E1} + \vec{E2} = \sigma /2 \epsilon _0 + \sigma /2 \epsilon _0 = \sigma /\epsilon _0$$
For glass, there's only the one charged surface.
I am not sure what you are referring to as being backwards.

$\sigma = Q/2A$ for Aluminum since the charge is distributed over both surfaces of the sheet. For glass $\sigma = Q/A$ since, as you point out, there is only one charged surface of area A.

Just substitute these values for $\sigma$ into your equation to get my answers.

AM

#### hhegab

Peace,

In Serway's Physics for S&E ,6th edition, chapter 24 section 4 (conducing surfaces), when using Gauss' Law
$$\oint{\vec{E} \cdot d\vec{A}} = \frac{Q_{enc}}{\epsilon_0}$$
is used, we have A as the TOTAL surface area not a surface of a particular region on the material under study. This is how I reached my answer, what do you think?

hhegab

#### Andrew Mason

Homework Helper
hhegab said:
Peace,

In Serway's Physics for S&E ,6th edition, chapter 24 section 4 (conducing surfaces), when using Gauss' Law
$$\oint{\vec{E} \cdot d\vec{A}} = \frac{Q_{enc}}{\epsilon_0}$$
is used, we have A as the TOTAL surface area not a surface of a particular region on the material under study. This is how I reached my answer, what do you think?

AM

#### hhegab

$$\sigma = \frac{Q}{A}$$ in both cases;

hhegab

#### Andrew Mason

Homework Helper
hhegab said:
$$\sigma = \frac{Q}{A}$$ in both cases;

hhegab
So how does that occur if the surface charge density of the top surface of the alluminum conductor is the same as the bottom surface and there is 0 field in between the two surfaces?

AM

#### hhegab

Peace!

The field lines are perpendicular to the surface, and directed either inward or outward according to the sign of sigma, when they added (vectirially) they will give a zero field.
Also, even in the presence of a field inside the conductor, when we put it an an external elecric field, the free charges will arrange themselves on its surface in such a configuration that after about 10^-16 sec. they will cancell the originl field.

hhegab