1. Consider a thin ring of radius R. The ring is uniformly charged with a total charge Q. Write an expression for the linear charge density i of the ring.
1. Consider a thin ring of radius R. The ring is uniformly charged with a total charge Q. Write an expression for the linear charge density i of the ring.
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Transcribed Image Text:Electric field of a ring of charge
1. Consider a thin ring of radius R. The ring is uniformly charged with a total charge Q.
Write an expression for the linear charge density à of the ring.
2. The ring has rotational symmetry. That means the object can be rotated about a
fixed axis without changing the overall shape. Which one is this axis? (Call it axis of
symmetry). Make a small sketch of the ring and its axis of symmetry.
Your goal for the next parts is to calculate the net electric field due to the charged
ring at a point P located at a distance z from the center of the ring along the axis of
symmetry of the ring.
3. Make a sketch of the ring and the point P. Imagine to divide the ring in many tiny
elements of charge dq. Inspecting the symmetry of the problem, what do you think
will be the direction of the net electric field at point P?
4. Consider an element of charge dq on the ring, write the magnitude of the electric
field due to this element of charge at point P. Introduce a coordinate system to
define the variables you may need.
5. Based on your answer to part 3. and therefore the symmetry of the problem, write
the electric field magnitude at point P due to the element of charge dq, after taking
into account of any cancellation (for example if only the y-component survives, or
only the x-component, and so multiply by the appropriate cosine or sine of the
appropriate angle).
6. Use Pythagoras' theorem and trigonometry to make sure your answer to the
previous part is in terms of: z, R, dq and €, only.
7. Integrate appropriately your expression in part 6. to add up the effect of all the dq
elements of charge on the ring. Your final expression should be in terms of: z, R, Q
and e, only.
8. Check with your textbook that your final answer is correct. Then make a plot of the
electric field magnitude E(z) as a function of z for z > 0.
9. What is the expression for the electric field due to the ring in the limit that the point
Pis very far away from the ring (z » R) ? How does the expression you found
compare with the electric field of a point charge?
Expert Solution
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Linear charge density of a conductor is the charge per unit length of the conductor. The ring may be supposed to be equivalent to a straight conductor of length equal to the perimeter of the ring.
The formula to calculate the perimeter of the ring is given by
Here, is the perimeter and is the radius of the ring.
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