Griffiths Electrodynamics book: Electric potential

[chingcx]

Homework Statement

This is from Prob. 2.25
Two point charges with separation d, P is a point at a distance z above the mid-point of the charges.
The last sentence asked if one of the positive charges is changed to a negative one, what is the potential at P? What field does it suggest? Explain the discrepancy.

Homework Equations

E = -grad V

The Attempt at a Solution

V=0 obviously
E = -grad V = 0
What is the reason behind that gives this result?

 

[gabbagabbahey]

Remember, $-\vec{\nabla}V=\frac{\partial V}{\partial x}\hat{x}+\frac{\partial V}{\partial y}\hat{y}+\frac{\partial V}{\partial z}\hat{z}$ Since you only know V on the z-axis, you cannot possibly calculate $\frac{\partial V}{\partial x}$ and $\frac{\partial V}{\partial y}}$. Clearly, any E-field will point in the x-direction, and so it is necessary to determine V(x,y,z) at points off of the z-axis to find $\frac{\partial V}{\partial x}$ and hence E.

 

[E&M]

I had exactly the same question and I kind of understand what you are saying but since V is scalar and E is vector, isn't V supposed to possesses all the information that the three components of E possess? Would you please suggest me how you would compute E = -$$\nabla V$$ in this case and compare it with calculation using Gauss's Law?

Thanks

 

[gabbagabbahey]

The equation $\vec{E}=-\vec{\nabla}V$ applies to V(x,y,z) (in Cartesian coordinates anyways). The potential you've calculated is actually V(0,0,z) (the -potential on the z-axis) and so you do not know how V varies with x or y, and you cannot use V(0,0,z) to compute E. If you wanted to compute E from the potential, then you would need to find the potential at a general point (x,y,z) (or even just a point on the x-axis in this case) first and use that potential.

In some cases, you know from symmetry that E points in the z direction and so knowing the functional dependence of V(z) is enough. In this case however, E points in the x direction and so you need to know the functional dependence of V(x).

 

[E&M]

Thanks,

I also have another question...
Two infinite parallel plates separated by a distance s are at potential 0 and V_0
a) Use Poisson's eqn to find potential V in the region between the plates where the space charge density is rho = rho_0(x/s). The distance x is measured from the plate at 0 potential.
b)What are the charge densities in the plate?

For this problem, I started with number of ways but none of them seem to be working.

 

[gabbagabbahey]

You should start a new thread for that problem.