A square nonconducting plate, 2a on a side, has a total charge Q uniformly spread over its surface. Calculate the electric field vector E at distance z above the plate's center. Use the electric field at a distance z above the midpoint of a line of charge of length 2L as a shortcut when solving.

E' = Qa/4πε[√(a² + z²)]³

Explanation:

Since the non-conducting plate is symmetric, a small charge element dq generates an electric field dE at a distance R from itself and a distance z above the center of the plate. Since the plate is symmetric, we only have the vertical component of the electric field acting at the center so dE' = dEcosθ where θ is the angle between R and the plate.

So, dE' = dEcosθ = dqcosθ/4πεR²

Let σ represent the surface charge density of the plate. So, for a small elemental area dA, dq = σdA.

Substituting this into dE' we have

dE' = σdAcosθ/4πεR²

Also cosθ = a/R where a is half the length of side of the plate of side length, 2a.

So, dE' = σdAa/4πεR³

Also R² = a² + z²

R = √(a² + z²)

So,  dE' = σdAa/4πε[√(a² + z²)]³

Now, dA = dxdy

dE' = σadxdy/4πε[√(a² + z²)]³

So, the total electric field at z is obtain by integrating dE'

E' = ∫dE' = ∫σadxdy/4πε[√(a² + z²)]³ = σa∫dxdy/4πε[√(a² + z²)]³

We integrate dx and dy from -a to a.

So,

E' = σa[2a][2a]/4πε[√(a² + z²)]³

E' = σa³/πε[√(a² + z²)]³

Since the total chare Q = σA where A is the are of the plate. A = (2a)² = 4a²

Q = σA = 4σa²

σ = Q/4a²

substituting σ into E', we have

E' = (Q/4a²)a³/πε[√(a² + z²)]³

E' = Qa/4πε[√(a² + z²)]³

elly's long hair:

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b. if she has overdraft protection, then there's no problem.   she can go ahead and write a check that her account can't cover, then sometime later, pay the bank some outrageous usury for loaning her a few bucks for a few days, but with beautiful hair.

c. if elly has some principles, and doesn't want to write a bad check, then now she has a problem.   her desire for the new hairstyle is in conflict with her honesty and integrity.

d.   if the bank will not process a check she writes, then even if she had a million $$ in her account, she couldn't use any of it.   the amount she has in her account is not the problem.   and if i may suggest, she might as well close her checking account, and put the money where it'll work for her.   it's not doing her a bit of good in a checking account, since she can't write any checks.

justin can't pay the full balance:

a. he can't even handle the credit limit he has now.   under no circumstances should he ask for a higher limit.

b.   he can ask for the grace period to be extended.   but that's no solution, the bank will almost certainly refuse anyway, and there are much better options.   he knew the rules of his credit card when he started using it.

c.   he definitely should pay the interest on the balance, plus as much of the balance as he can cover at the same time.

d.   he definitely should pay the minimum otherwise they could shut down his card.   he should also pay as much more as he possibly can.   he should also remember the awkward situation he's in, stop buying stuff on credit that he can't afford, and avoid adding any more debt.

i'd say that justin's best best option isn't on this list of choices.   what he has to do:   pay as much as he can at least the minimum payment plus as much more as he can and then avoid adding more debt.

reserve is the answer