Problem 1: Electric field on the axis of a charged rod 1) Draw and label an infinitesimal charge dq 1 on the rod (don't place it at a special...

Problem 1:

Electric field on the axis of a charged rod

1)    Draw and label an infinitesimal charge dq1 on the rod (don't place it at a special location, i.e., not at the ends or in the middle). Draw the electric field  at P due to dq1.

2)    What is the formula for ? Make clear on the diagram any symbols you use in your formula. 

3)    What property of the charge makes your formula valid?

4)    Now draw another infinitesimal charge dq2 on the diagram above and the corresponding electric field . Be qualitatively correct when drawing the length of .

5)    Use vector addition to draw the electric field due to the two infinitesimal charges dq1 and dq2.

6)    Express the electric field due to the two infinitesimal charges dq1 and dq2.

If we divide the rod into many pieces, each small enough to be considered a point charge, the electric field created by the rod would be:

Since all 's have the same direction, the total electric field  is given by .

To define the integral fully, introduce an x-axis with its origin at the left end of the rod. Consider the infinitesimal charge dq in the infinitesimal length dx of the rod between x and x+dx.

7)    What is the distance between the charge dq and point P? 

8)    If lambda is the charge per unit length, how can you express dq in terms of lambda and dx.

9)    Write dE in terms of constants, x, and dx.

10)  Complete the expression for the E field due to the entire rod below

11)  Compute the above integral

12)  

a) Reexpress the above expression in terms of the total charge -q of the rod.

b) If point P is far away from the rod, what expression do you expect for E in terms of -q and a (it is not infinite!). Verify your answer by simplifying your expression for a >> L.