How to Prove E and H Satisfy Maxwell's Equations?

[likephysics]

Homework Statement

I am trying to solve prob 4.107 in Schaums' Vector analysis book.

Show that solution to Maxwells equations -

$$\Delta$$xH=1/c dE/dt, $$\Delta$$xE= -1/c dH/dt, $$\Delta$$.H=0, $$\Delta$$.E= 4pi$$\rho$$
where $$\rho$$ is a function of x,y,z and c is the velocity of light, assumed constant, are given by

E = -$$\Delta$$$$\phi$$-1/c dE/dt, H= $$\Delta$$xA

where A and $$\phi$$, called the vector and scalar potentials, respectively satisfy the equations
$$\Delta$$.A + 1/c d$$\phi$$/dt =0
$$\Delta$$^2 $$\phi$$ - 1/c (d^2 $$\phi$$/dt^2) = -4pi$$\rho$$
$$\Delta$$^2 A = 1/c^2 (d^2A/dt^2)

Homework Equations

The Attempt at a Solution

I don't understand the problem. Should I show that E = -$$\Delta$$$$\phi$$-1/c dE/dt, H= $$\Delta$$xA satisfies the vector and scalar potential equations?

 

[tiny-tim]

Hi likephysics!

(have a delta: ∆ and a phi: φ and try using the X² tag just above the Reply box )

Should I show that E = -$$\Delta$$$$\phi$$-1/c dE/dt, H= $$\Delta$$xA satisfies the vector and scalar potential equations?

No, you should assume that the A,φ equations are satisfied, and then prove that E and H (derived from A and φ) satisfy Maxwell's equations.

 

[gabbagabbahey]

No, you should assume that the A,φ equations are satisfied, and then prove that E and H (derived from A and φ) satisfy Maxwell's equations.

I disagree, that seems to be proving the reverse statement of what the problem statement asks for.

I would assume that E and H satisfy Maxwell's equations (so that they are solutions to said equations, as per the first premise of the problem statement), then substitute in your expressions for them in terms of the vector and scalar potentials (the second premise of the problem statement) and use appropriate vector product rules to show that Maxwell's equations, in terms of the potentials, reduce to the final 3 equations you are given (the intended conclusion).