e&m 1

right hand rule for cross products A x B = C

A - index finger

B - middle finger

C - thumb

the curl of a gradient

is always 0

the divergence of curl

is always 0

Stokes’ theorem says…

the integral of curl is a path integral

Green’s theorem says…

the integral of divergence is a surface integral

what is the flux through any surface enclosing a charge?

$$\frac{q}{\epsilon_0}$$

E$\cdot$ da for a sphere

= E(da) because E and da are parallel

radius bounds for E outside of a surface?

r > R

radius bounds for E inside of a surface?

r < R

charge density $\rho$ for a uniformly charged surface

a constant: total charge / volume or total charge / surface area or total charge / length

charge density $/rho$ for a non-uniformly charged surface

function: total charge / volume or total charge / surface area or total charge / length

where r is of the Gaussian surface/is allowed to vary

the electric potential V inside of a closed solid surface

V=-\int_{\infty}^{surface}\!E_{outside}\cdot dl-\int_{surface}^{t\arg et}E_{\imaginaryI nside}\cdot dl

some properties of conductors

1. they have equipotential throughout the entire object

2. their E fields are normal to the surface when close to the object

3. their internal E field is 0 regardless of shape, shell, or solid

4. all of their net charge lies on the surface

5. they have 1+ free electrons per atom

6. their inner charge density is 0

what are some surfaces where E$/cdot$dA can be simplified to E(dA)?

surfaces where E is parallel to the normal vectors dA at every point;

common examples are

1. spheres

2. cylinders

3. infinite flat sheets (“pillboxes”)