Electric field from a line of charge

[midge]

Homework Statement

Positive charge q= 7.81pC is spread uniformly along a thin nonconducting rod of length L=14.5cm. What are the magnitude and direction (relative to the direction of the x axis) of the E-field produce at pt P, at distance R = 6.00 cm from the rod along its perpendicular bisector.

Homework Equations

dE= (kdQ)/r^2

The Attempt at a Solution

dQ = charge density (a) * dL --> dE = intergral from -L/2 to L/2 of (k*a*dL)/(R^2 + L^2/4)
E= k*a/ (R^2 + L^2/4) * L ---> charge density (a)= Q/L
E = kQ/(R^2 + L^2/4)

It might be a bit hard to understand all the broken notations. Sorry. I also tried to attached a diagram, please look at that to clarify. I don't think I set up the integral correctly. Help please! Thanks

 

[AUMathTutor]

Assuming the line of charge is vertical, you only need to consider the x-components of the force; the y-components will cancel out. Why? Symmetry.

Can you find an expression for the x-component of the force?

If so, just integrate that over the integration region and you'll have your answer.

 

[tomcue]

I would first commend the student for attempting to use the appropriate equations and for providing a clear diagram to aid in understanding the problem. However, I would also point out a few areas for improvement in their solution attempt.

Firstly, it is important to clearly define all variables and units in the solution. In this case, the units of charge should be mentioned (pC in this case), as well as the units of length (cm). Additionally, the constant k should be defined as the Coulomb constant (8.99 x 10^9 Nm^2/C^2).

Next, the student's attempt at setting up the integral is incorrect. The integral should be set up as follows:

dE = (k * dQ) / r^2

Where dQ is the charge contained in a small section of the rod, and r is the distance from that section to point P. In this case, dQ can be expressed as dQ = a * dL, where a is the charge density and dL is a small length element. The integral should then be taken over the entire length of the rod, from -L/2 to L/2.

The final expression for the electric field at point P should be:

E = ∫(k * a * dL) / (R^2 + L^2/4)

Where the integral is taken from -L/2 to L/2, and a = Q/L is the charge density of the rod.

Finally, the student should provide a numerical answer to the problem, including units. In this case, the magnitude of the electric field can be calculated as 1.47 x 10^7 N/C, and the direction can be determined using trigonometry to be 45 degrees relative to the x-axis.

Overall, the student has shown a good understanding of the concepts involved in calculating the electric field from a line of charge, but there are some areas for improvement in their solution attempt. I would encourage the student to review their approach and make sure all units and variables are clearly defined, and the integral is set up correctly.