Field on a plane from line charge

In summary: And since the line is moving, the field at X and Y is going to change from one time step to the next.In summary, the student attempted to find the field at a point on a plane and found that the integral satisfied both conditions for a small and large x,y.
  • #1
bowlbase
146
2

Homework Statement


A wire at extends for -L to L on the z-axis with charge ##\lambda##. Find the field at points on the xy-plane


Homework Equations



##E(r)=k\int\frac{\rho}{r^2}dq##
##k=\frac{1}{4\pi ε_o}##

The Attempt at a Solution



First time I've looked for field on a plane so I wasn't sure if I'm doing this correctly.

##dq=\lambda dz=2L\lambda##
I made a right triangle with one side L and the other two x and y.
##r^2=L^2+x^2+y^2## where only x and y change so I have dxdy.

So, the final integral

##k(2L\lambda)\int\int\frac{1}{L^2+x^2+y^2}dxdy## with limits 0→∞.

I expect at large x,y that the field should be very small and small x,y it should be large. This integral satisfies both of those conditions.

The method I saw our professor do for another, somewhat similar, problem was exceedingly more complicated than this.
 
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  • #2
Hmm, the question is to find the field at some point (x, y, 0). So you should fix x and y, and add up all the contributions from the charges on the z-axis.
Therefore I would expect something like
$$E(x, y, 0) = k \int_{-L}^L \frac{\rho}{r^2} \, dz$$
(where did your ##\rho## go? Should there not be something with ##\lambda## in there)?

Try drawing such a point, and calculating ##r## and ##\rho \, dz## from the geometry of the problem rather than by trying to copy your notes.
 
  • #3
##\rho## is the charge density ##\lambda##
I was trying to use ##\rho## in the general sense but I sort of got ahead of myself putting it in there.

##E(r)=k\int\frac{1}{r^2}dq##

##dq=\lambda dz##

so, ##E(r)=k\int\frac{\lambda}{r^2}dz##


As soon as I walked away from the computer I knew I was wrong. My whole way of thinking was stuck in previous problems I've done.

X and Y are my points of interest and Z is my variable (for the line charge not on the plane). So if I want to figure out what the field is at X and Y I need to count up all the little pieces of charge at distance 'r' from them to the line.

I have ##\int_{-L}^{L} \frac{\lambda}{z^2+x^2+y^2}dz##
 

1. What is a line charge?

A line charge is a theoretical concept used in physics to represent a one-dimensional distribution of electric charge along a straight line. It is often used to model the behavior of electric currents or charged wires.

2. How is the field on a plane from a line charge calculated?

The field on a plane from a line charge is calculated using the Coulomb's law, which states that the electric field at a point is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance between the point and the charge.

3. What factors affect the magnitude and direction of the field on a plane from a line charge?

The magnitude and direction of the field on a plane from a line charge are affected by the distance from the line charge, the magnitude of the charge, and the angle at which the plane intersects the line charge.

4. Can the field on a plane from a line charge be negative?

Yes, the field on a plane from a line charge can be negative. This occurs when the plane is located at a point where the electric field is directed away from the line charge, which happens when the plane is outside of the line charge.

5. How is a line charge different from a point charge?

A line charge is a one-dimensional distribution of charge along a straight line, while a point charge is a single point with a finite amount of charge. The field on a plane from a line charge is more complex than the field from a point charge, as the distribution of charge along a line can vary and affect the field in different ways.

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