Electric field above the centre of a 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?