A thin-walled hollow circular glass tube, open at both ends, has a radius R and

Question

A thin-walled hollow circular glass tube, open at both ends, has a radius R and length L (see the figure). The axis of the tube lies along the x axis, with the left end at the origin. The outer sides are rubbed with silk and acquire a net positive charge Q distributed uniformly. Determine the electric field at a location on the x axis, a distance w from the origin. Carry out all steps including checking your result. Explain each step. You may have to refer to a table of integrals. (Do this on paper. Your instructor may ask you to turn in this work.)



Step 1:
Since we know the electric field of a charged ring, divide the tube into rings, of thickness dx. Consider a representative ring somewhere in the middle of the tube, with its center at <x,0,0>. Draw a diagram illustrating this situation.

Step 2:
How much charge dQ is on this ring? Write your answer symbolically. (Use any variable or symbol stated above as necessary.)
dQ =

.

What is the distance from this ring to the observation location?
d =

.

What is the vector from source to observation location?
=

.

What is the integration variable? (Use any variable or symbol stated above as necessary.)


.
What is the lower integration limit? (Use any variable or symbol stated above as necessary.)


.
What is the upper integration limit? (Use any variable or symbol stated above as necessary.)


.
Step 3.
Evaluate the integral, using the tool of your choice.

Step 4.
Check the units, and the special case where w >> R.

Explanation / Answer

Below will be the answers I got: Step 2: How much charge dQ is on this ring? Write your answer symbolically. (Use any variable or symbol stated above as necessary.) dQ =Qdx/L . What is the distance from this ring to the observation location? d = w-x . What is the vector from source to observation location? = . What is the integration variable? (Use any variable or symbol stated above as necessary.) x . What is the lower integration limit? (Use any variable or symbol stated above as necessary.) 0 . What is the upper integration limit? (Use any variable or symbol stated above as necessary.) L . Step 3. Evaluate the integral, using the tool of your choice.

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