[symbolic] Linear uniform charge density (E field at a point)

In summary, the problem involves finding the electric field at point P, located at (0,y), due to a rod of length L centered at (0,0). The solution involves integrating from -L/2 to L/2 with respect to x, where x is the distance along the horizontal axis, and using trigonometric substitution to simplify the resulting integral.
  • #1
syntroniks
3
0

Homework Statement


Center a rod of length L at (0,0) with the length going horizontally.

Take a point P at (0,y).

Find the electric field at P.

Homework Equations


[tex]\lambda[/tex]=Q/L
[tex]E= \int k*dQ/R^{2}[/tex]

The Attempt at a Solution


I am integrating from -L/2 to L/2
Since Q=lambda*L, I guess differentially dQ=lambda*dL.

Substituting that into the integral, it becomes:
[tex]k*\lambda \int dL/R^{2}[/tex]
from -L/2 to L/2 of course.

R is pretty messy so I'll just write what I came up with for [tex]R^{2}[/tex]:
[tex]R^{2}=(L^{2}/4)+y^{2}[/tex]

So... Doesn't this seem pretty reasonable? I just want to be double sure that this is OK.
 
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  • #2
There may be a confusion with L, used in two different ways here.
I suggest you change dQ=lambda*dL to dQ=lambda*dx, where x is a distance along the x axis. This x runs from -L/2 to L/2.
I think you'll find that R² = x² + y².
Looks like one of those trig substitution integrals.
 
  • #3
Thankfully for this particular problem I get to do the integration by software... kinda. Turns out to involve arctangent and a relative mess of symbols.

Thanks for the suggestion about variables, it is definitely more clear that way.
 
  • #4
Looks like the substitution x = y*tan A really simplifies it!
And the A is a real angle in the problem.
 

FAQ: [symbolic] Linear uniform charge density (E field at a point)

1. What is the symbolic representation of linear uniform charge density?

The symbolic representation of linear uniform charge density is λ.

2. How is linear uniform charge density calculated?

Linear uniform charge density is calculated by dividing the total charge by the length of the charged object.

3. What is the unit of linear uniform charge density?

The unit of linear uniform charge density is Coulombs per meter (C/m).

4. How does linear uniform charge density affect the electric field at a point?

The linear uniform charge density is directly proportional to the electric field at a point. As the charge density increases, the electric field also increases.

5. Can linear uniform charge density be negative?

Yes, linear uniform charge density can be negative if the charged object has a net negative charge. However, the magnitude of the charge density is always positive.

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