Integrating a charge along a line

In summary, the problem involves finding the voltage (V) at a point P on the y-axis due to a line of charge with a total charge Q = 200[nC] spread along a line from x=−100[mm] to x=+100[mm]. Using the equation dV = k*dq/r, where dq = λ*dx and r = sqrt(x^2 + y^2), we can integrate along the line of charge to find V as a function of y. Taking the derivative with respect to y, we can find the component of the electric field along the y-axis. And finally, by taking the derivative with respect to y again, we can determine the location on the y-axis where the
  • #1
burns12
14
0

Homework Statement



total charge Q = 200[nC] is spread along a line from x=−100[mm] to x=+100[mm] .
a) write the dV caused by each dQ=λ dx .
b) integrate along the line of charge ; write V as a function of y .
c) take the derivative with y , to obtain that component of the Electric (vector) field .
d) take the derivative with y , again , to find where (along y-axis) the Ey is maximum


Homework Equations


We've used potential energy U=1/2QV...V=kQ/r...that's about it.


The Attempt at a Solution


What I can't get started with is the writing dV caused by each dQ. I'm not even sure what that means. And how can I write V as a function of y and take its derivative if there are no y's in any of these equations?

Any guidance here would be awesome, thanks
 
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  • #2
hi burns12, welcome to PF.
Take a small element dx at a distance x from the origin.
Consider a point P at a distance y from the origin along y-axis.
Charge dq in the dx element is λ*dx, where λ = Q/L.
As you have mentioned
dV = k*dq/r. Put the value of dq and r.
To find V at P, take the integration between the limits x = -0.1 m to 0.1 m.
 
  • #3
Thanks!

Ok so, the y comes into play as a distance on the y-axis that this dq charge is going to affect?
And do I integrate the dV = k*dq/r ? Or do I sub in λdx for dq and then integrate with respect to x to find dV?
I'm still not sure where this distance y would come into this.
 
  • #4
burns12 said:
Thanks!

Ok so, the y comes into play as a distance on the y-axis that this dq charge is going to affect?
And do I integrate the dV = k*dq/r ? Or do I sub in λdx for dq and then integrate with respect to x to find dV?
I'm still not sure where this distance y would come into this.
dq = λ*dx and r = sqrt( x^2 + y^2)
 
  • #5
Alright awesome I think I got it. Is the r = to sqrt(x^2 +y^2) because it's like the components of r in a sense?
 
  • #6
burns12 said:
Alright awesome I think I got it. Is the r = to sqrt(x^2 +y^2) because it's like the components of r in a sense?
Yes.
 
  • #7
Awesome, thanks for your help, that had me stumped for 2 days now ha
 

1. What is the concept of integrating a charge along a line?

Integrating a charge along a line is a mathematical technique used to calculate the total electric charge distribution along a specific path or line. It involves summing up the infinitesimal charge elements along the line to obtain the total charge.

2. How is the line of integration chosen in this process?

The line of integration is typically chosen based on the symmetry of the charge distribution. It can also be chosen to simplify the calculation or to align with known boundaries or axis.

3. What are the steps involved in integrating a charge along a line?

The first step is to choose the line of integration and determine its limits. Then, the charge distribution function along the line is expressed as a function of the variable of integration. The next step is to integrate this function over the chosen limits, taking into account any constants or variables. Finally, the result is interpreted in terms of the total charge distribution along the line.

4. Can this technique be applied to any charge distribution?

Yes, integrating a charge along a line can be applied to any continuous charge distribution, as long as the line of integration is chosen appropriately and the integral can be evaluated.

5. What are some practical applications of integrating a charge along a line?

Integrating a charge along a line is commonly used in electromagnetism to calculate the electric field or potential along a specific path. It is also used in antenna design and analysis, as well as in calculating the forces between charged particles along a line. It has numerous other applications in various fields of physics and engineering.

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