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19 Nov 2019

# 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 tubelies along the x axis, with the left end at the origin. The outersides are rubbed with silk and acquire a net positive charge Qdistributed uniformly. Determine the electric field at a locationon the x axis, a distance w from the origin. Carry out all stepsincluding checking your result. Explain each step. You may have torefer to a table of integrals.

Step 1: Since we know the electric field of a charged ring, divide thetube into rings, of thickness *dx*. Consider arepresentative ring somewhere in the middle of the tube, with itscenter at . Draw a diagram illustrating this situation. Step 2: (a) How much charge *dQ* is on this ring? Write youranswer symbolically. (Use any variable or symbol stated above asnecessary.) dQ = ________ (b) What is the distance from this ring to the observationlocation? d = ________ (c) What is the vector from the source to observationlocation? r = ________ (this answer is a vector) (d) What is the integration variable? (Use any variable orsymbol stated above as necessary.) answer = ________ (e) What is the lower integration limit? (Use any variable orsymbol stated above as necessary.) answer = ________ (f) What is the upper integration limit? (Use any variable orsymbol stated above as necessary.) answer = ________ Step 3: Evaluate the integral, using the tool of your choice. Step 4: Check the units, and the special case where w>>R.

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 tubelies along the x axis, with the left end at the origin. The outersides are rubbed with silk and acquire a net positive charge Qdistributed uniformly. Determine the electric field at a locationon the x axis, a distance w from the origin. Carry out all stepsincluding checking your result. Explain each step. You may have torefer to a table of integrals.

Step 1:

Since we know the electric field of a charged ring, divide thetube into rings, of thickness

*dx*. Consider arepresentative ring somewhere in the middle of the tube, with itscenter at . Draw a diagram illustrating this situation.Step 2:

(a) How much charge

*dQ*is on this ring? Write youranswer symbolically. (Use any variable or symbol stated above asnecessary.)dQ = ________

(b) What is the distance from this ring to the observationlocation?

d = ________

(c) What is the vector from the source to observationlocation?

r = ________ (this answer is a vector)

(d) What is the integration variable? (Use any variable orsymbol stated above as necessary.)

answer = ________

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

answer = ________

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

answer = ________

Step 3:

Evaluate the integral, using the tool of your choice.

Step 4:

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