- #1

Fledy

- 2

- 0

So then.

I also think I have a fairly good grasp of multivariable calc (i'm taking a correspondence course). So of course, even though we don't deal with a full-fledged surface integral in AP Physics C, I understand the formulation of Gauss's law in the terms of a surface integral.

With that in mind, I have been trying to confirm Gauss's law for a point charge and a spherical gaussian surface centered at the charge using multivariable calculus techniques. I know, it defeats the purpose of Gauss's law, but I like to see these things. Problem is, I wind up finding an integral that evaluates to 0. Ahem.

Here's what I first did: (I won't use the Tex notation for now, but I'll use boldface for vectors)

Position vector

**x**= x

**i**+ y

**j**+ z

**k**

**E**(

**x**) = k q

**x**/ x³

where x is just the magnitude of

**x**.

Next, I parameterized a sphere with radius r as

**A**(u,v) = r (sin(u) cos(v), sin(u) sin(v), cos(u)).

(u runs from 0 to pi, v runs from -pi to pi)

Now, to evaluate the integral of

**E**.

**dA**, I had to have an expression for

**dA**. I used the differential area element. This is given by

**dA**= (

**∂A/∂u**×

**∂A/∂v**) du dv

Finally, I integrate:

**E(A(**u,v)) . (

**∂A/∂u**×

**∂A/∂v**) du dv

with u running from 0 to pi, and v running from -pi to pi.

Problem: This whole integral evaluates to 0, and not the expected q / epsilon naught.

...

Okay...

Time for a different approach! If I take the divergence of

**E**(x,y,z), I should get rho / epsilon nought, the charge density over epsilon nought.

I basically used the same definition of electric field above, expanding r and

**r**in terms of x, y, and z. Simply put, replace

**r**with (x, y, z) and r with sqrt(x² + y² + z²). Del = (∂/∂x, ∂/∂y, ∂/∂z).

Some basic rules of calculus let me calculate

**del**.

**E**

= ∂E/∂x + ∂E/∂y + ∂E/∂z

...

And it came out to 0. Again. ... Oops.

So basically, I want to know what I'm doing wrong. Why doesn't doing this yield the net charge over epsilon nought, or the charge density over epsilon nought as they supposedly should? Is this an incorrect application of multivariable calculus or gaussian surfaces? I really must know where I went wrong.

Thanks

Much love,

Steve