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Griffiths Electrodynamics book: Electric potential

  1. Oct 12, 2008 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations
    E = -grad V


    3. The attempt at a solution
    V=0 obviously
    E = -grad V = 0
    What is the reason behind that gives this result?
     
  2. jcsd
  3. Oct 12, 2008 #2

    gabbagabbahey

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    Remember, [itex]-\vec{\nabla}V=\frac{\partial V}{\partial x}\hat{x}+\frac{\partial V}{\partial y}\hat{y}+\frac{\partial V}{\partial z}\hat{z}[/itex] Since you only know V on the z-axis, you cannot possibly calculate [itex]\frac{\partial V}{\partial x}[/itex] and [itex]\frac{\partial V}{\partial y}}[/itex]. 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 [itex]\frac{\partial V}{\partial x}[/itex] and hence E.
     
    Last edited: Oct 12, 2008
  4. Oct 13, 2008 #3

    E&M

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    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 possess all the information that the three components of E possess? Would you please suggest me how you would compute E = -[tex]\nabla V [/tex] in this case and compare it with calculation using Gauss's Law?

    Thanks
     
  5. Oct 13, 2008 #4

    gabbagabbahey

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    The equation [itex]\vec{E}=-\vec{\nabla}V[/itex] 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).
     
  6. Oct 13, 2008 #5

    E&M

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    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.
     
  7. Oct 13, 2008 #6

    gabbagabbahey

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    You should start a new thread for that problem.
     
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