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Voltage Difference Along a Path

  1. Feb 13, 2016 #1
    1. The problem statement, all variables and given/known data

    Given the electric field E = (2x−y2)ax + (3z−2xy)ay + 3yaz, and the piecewise linear path joining the points A(−2,1,−1), P(2,1,−1), Q(2,3,−1) and B(2,3,1), find −∫E⋅dl from A to P − ∫E⋅dl from P to Q − ∫E⋅dl from Q to B along the straight line segments.

    2. Relevant equations

    −∫E⋅dl

    (x - x1) / (x1 - x2 ) = (y - y1) / (y1 - y2 ) =(z - z1) / (z1 - z2 )

    3. The attempt at a solution

    I have solved this problem going direct from point a to b by using the above equation and found that the answer is 8 v. This is confirmed by the answer key. Here is my problem, when I set up the line equation to go from point A to P I get the following;

    (x + 2) / (-2 - 2) = (y - 1) / (1 - 1) = (z + 1) / (-1 + 1)

    or, (x + 2) / -4 = (y - 1) / 0 = (z + 1) / 0

    I am ultimatley going to solve for y and z in terms of x to express my integral in terms of just x and I know how to get rid of the pesky dy and dz as well but I need the line equation to not blow up on me.

    How do I resolve the division by 0? I know that the VAQ = 4 v from the solution but I am not sure what to do about the line equation.

    I have an exam on Monday and I am certain a question like this will appear.
     
    Last edited: Feb 13, 2016
  2. jcsd
  3. Feb 13, 2016 #2

    TSny

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    Gold Member

    Did you mean to write A to P?
    The division by 0 comes from the special nature of the path from A to P. For that path, how would you simplify the expression

    ##\vec{E}\cdot d\vec{l} = E_xdl_x+E_ydl_y+E_zdl_z##?
     
  4. Feb 13, 2016 #3
    I did mean to write A to P. I fixed it.

    if y = z then dly = dlz = 0

    so the expression simplifies to ##\vec{E}\cdot d\vec{l} = E_xdl_x##
     
  5. Feb 13, 2016 #4
    If this is the case then I have
    -∫ (2x - y^2) dx from x= -2 to 2 but Y =1

    so, -∫ (2x - 1^2) dx = 4

    This is the result I am looking for. I want to be sure I didnt luck in to it and that I simplified it correctly as you suggested.
     
  6. Feb 13, 2016 #5

    TSny

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    That looks very good.
     
  7. Feb 13, 2016 #6
    Awesome, Thanks for your help
     
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