Electric field above the centre of a rectangle

[haruspex]

Writing $y=r \tan(\theta)$ and c=a/r:
$\int \frac{rady}{(y^2+r^2)(y^2+r^2+a^2)^\frac 12}=\int \frac{c.d\theta}{(c^2+\sec^2(\theta))^\frac 12}$
$=\int \frac {c.d\sin(\theta)}{(1+c^2\cos^2(\theta))^\frac 12}$
writing s for sin(theta) and f² for a²/(a²+r²):
$=\int \frac{f ds}{(1-f^2s^2)^\frac 12}$
Writing s=sin(φ)/f:
$=\int d\phi=[\phi]$
Then it is just a matter of unwrapping all the substitutions.

 

[TSny]

Ez = 102149.75 N/C

As a rough check of your numerical answer you can compare your answer to what you would get if you lumped all the charge of the plate together as a point charge at the origin.

 

[says]

As a rough check of your numerical answer you can compare your answer to what you would get if you lumped all the charge of the plate together as a point charge at the origin.

I'm not sure I follow?

 

[TSny]

I'm not sure I follow?

If you compressed all the charge of the plate into a point charge at the origin, what would be the magnitude of the E field at the point on the z axis at z = 10 cm?

Would you expect this value to be larger or smaller than the answer you got for the plate?

 

[says]

It would be the same wouldn't it?

I'm not sure how to compress all of the charge of the plate into a point source at the origin though.

 

[TSny]

It would be the same wouldn't it?

I'm not sure how to compress all of the charge of the plate into a point source at the origin though.

You can just imagine that you have a point charge at the origin that has a charge equal to the total charge of the plate.

(1) All the charge in the point charge would be a distance of 10 cm from the point on the z axis at z = 10 cm. Compare that to having the charge spread out on the rectangle. Is all the charge on the rectangle at a distance of 10 cm from the point on the z axis or is most of the charge on the rectangle farther away?

(2) Also, for the rectangle, you had to project out the z-component of the field of each small region of the rectangle. For the point charge, all of the charge produces E field entirely in the z-direction.

Considering (1) and (2), would you expect the field of the plate to be smaller or larger than the field of the point charge?

It is very easy to calculate the E field of the imaginary point charge. So, you can see if your expectation is verified.

 

[says]

1) if the charge is spread out then most of it is further away.

2) field of the plate would be smaller than that of the point charge

 

[TSny]

1) if the charge is spread out then most of it is further away.

2) field of the plate would be smaller than that of the point charge

Yes. Good. Did you calculate E for the point charge and compare it what you got for the plate?

 

[says]

Charge density = Q/A
4*10^-6 = Q/(0.10*0.20)
Q = 4*10^-6 / (0.10*0.20)
Q = 8*10^-8
E = kQ/r²
k=8.98*10^-9
r=0.10
E=718.4/0.01
E=71840 N/C

 

[TSny]

Charge density = Q/A
4*10^-6 = Q/(0.10*0.20)
Q = 4*10^-6 / (0.10*0.20)
Q = 8*10^-8
E = kQ/r²
k=8.98*10^-9
r=0.10
E=718.4/0.01
E=71840 N/C

Looks good. So, what does this tell you about your answer for the field of the rectangle?

 

[says]

That my answer is unreasonable because it is greater than that of the point charge calculation

 

[TSny]

That my answer is unreasonable because it is greater than that of the point charge calculation

Yes, there must be a mistake in your calculation for the rectangle. So, review the calculation to see if you can find an error. Could be a minor mistake somewhere.

Generally, it is a good idea to work the problem in symbols and wait until the end to plug in numbers.

 

[says]

Ez = ∫ 2kλr / (y²+r²) [ 0.10 / (y²+r²+0.10²)^1/2 dy

k = 8.98*10⁹
λ = 4*10^-6
r = 0.10

Ez = ∫ 7184 / (y²+0.10²) [ 0.10 / (y²+0.10²+0.10²)^1/2 dy

Ez = ∫ 718.4 / (y²+0.01)(y²+0.02)^1/2 dy

Ez = 71840tan^-1(7.07107y / √(50y²+1))

substituting y=0.05 into the equation

Ez = 102149.75 N/C

I think of all the steps this may have been where the error is... Ez = ∫ 718.4 / (y²+0.01)(y²+0.02)^1/2

 

[TSny]

dEz = (kλdx / r²) z/r
Ez = 2kλz ∫ dx / (z² + x²)^3/2

where
z = √(y²+r²)

Ez = 2kλ/z[ x / (y²+r²) + x²)^1/2] (bounds of integration are 0 and 0.10m)

Ez = 2kλr / (y² + r²) [ 0.10 / ((y²+r²)+0.10²)^1/2]

k = 8.98*10⁹
λ = 4.0*10^-6
r = 0.10

Ez = 7184 / (y² + 0.10²) [ 0.10 / ((y²+0.10²)+0.10²)^1/2]

Ez = [7184 / (y² + 0.01)] [ 0.10 / ((y²+0.02)^1/2]

Ez = ∫ [7184 / (y² + 0.01)] [ 0.10 / ((y²+0.02)^1/2] dy

Ez = 71840tan^-1 [ 7.07107y / √(50y²+1)

Ez = 102149.75 N/C

I'm not sure, but it looks to me like you might have a mistake in plugging in the limits of y at the very end. You might check this.

 

[says]

The limits of y are 0.05m and 0 though

 

[TSny]

Ez = 71840tan^-1(7.07107y / √(50y²+1))

I think this is OK.

substituting y=0.05 into the equation

Did you take care of both the upper and lower limits of y?

Ez = 102149.75 N/C

I don't get this answer when using your expression above with the limits for y.

 

[TSny]

The limits of y are 0.05m and 0 though

Aren't the limits -0.05 m and + 0.05 m? Or did you replace the lower limit by 0 and include a factor of 2?

 

[says]

Ez = 2kλr / (y² + r²) [ 0.10 / ((y²+r²)+0.10²)1]

I replaced the lower limit with 0 and included a factor of 2. :)

 

[TSny]

I replaced the lower limit with 0 and included a factor of 2. :)

OK So, you would have

Ez = (2) 71840tan^-1(7.07107(.05) / √(50(.05)²+1))

I don't get your answer when I evaluate this.

 

[says]

OK So, you would have

Ez = (2) 71840tan^-1(7.07107(.05) / √(50(.05)²+1))

I don't get your answer when I evaluate this.

I get 40378 N/C. Which is more reasonable.

If I've substituted 2 into the equation earlier though:

Ez = 2kλr / (y² + r²) [ 0.10 / ((y²+r²)+0.10²)^1/2]

Should I be substituting 2 into the equation:

Ez = (2) 71840tan^-1(7.07107(.05) / √(50(.05)²+1))

Or just leaving it as:

71840tan^-1(7.07107(.05) / √(50(.05)²+1))

 

[TSny]

You need to include the 2. This is the factor of 2 that comes from taking the lower limit to be 0.

Your answer of 40378 looks much better, but it is still different than what I'm getting. I get the argument of the inverse tangent function to be essentially 1/3. Do you agree?

 

[says]

Yes. It's ~0.28

 

[TSny]

Yes. It's ~0.28

I'm getting 0.3333, not 0.28.

 

[says]

7.07107(.05) / √(50(.05)²+1)

0.3535535 / 1.125 = 0.3142

Closer. I'm not sure what happened with my first calculation...

 

[TSny]

7.07107(.05) / √(50(.05)²+1)

0.3535535 / 1.125 = 0.3142

Closer. I'm not sure what happened with my first calculation...

The 1.125 is under a square root.

 

[TSny]

If I've substituted 2 into the equation earlier though:

Ez = 2kλr / (y² + r²) [ 0.10 / ((y²+r²)+0.10²)^1/2]

Should I be substituting 2 into the equation:

Ez = (2) 71840tan^-1(7.07107(.05) / √(50(.05)²+1))

Or just leaving it as:

71840tan^-1(7.07107(.05) / √(50(.05)²+1))

You did two integrations in which you replaced the lower limit with 0. So, overall, there will be two factors of 2.

 

[says]

Ok, I get this now:

2*71840tan^-1(0.333)
=46186 N/C

 

[TSny]

OK, good.

If you are expected to round off to an appropriate number of significant figures, I'll let you think about that. But I agree with your answer now.

 

[says]

How come the E field of an infinite sheet is much larger than the answer we calculated here? This perplexes me

E = λ / 2ε₀

where
λ:charge density

E = 4*10^-6 / [(2)(8.85*10^-12)]
E = 225988 N/C

 

[haruspex]

How come the E field of an infinite sheet is much larger than the answer we calculated here? This perplexes me

E = λ / 2ε₀

where
λ:charge density

E = 4*10^-6 / [(2)(8.85*10^-12)]
E = 225988 N/C

An infinite sheet at the same charge density is a lot more total charge!
OK, it's not as simple as that, but consider tiling the plane with 10cm x 20cm rectangles.
The point P is 10 cm from the sheet, and the centres of the next rectangles are, in one direction, only 40% further away. As you add in more surrounding tiles, is it that surprising that the total Ez field gets up to nearly 5 times as much?