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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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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?
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?
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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

C=2πR................................(1)

Here, C is the perimeter and R is the radius of the ring.

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