Electric field above the centre of a rectangle

[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?