Question

A long straight wire has a hollow spherical conductor of radius R hanging from its end. The wire carries a downward current I. You are curious about the magnetic field, if any, that might exist just outside the "equator" of the sphere, i.e. the circle created by the intersection of the surface of the ball with a horizontal plane through its center, shown by the solid line a) What is the amount of charge on the sphere as a function of time? You are free to choose the initial amount of charge o (Of course if the charge isn't obvious you can start with the definition of current I and figure it out) b) What is the electric field just outside the sphere as a function of time in terms of I and Qo? (Of course if you don't remember a simple answer, you could use Gauss's law to figure it out). i) Draw a few electric field vectors around the sphere to help you visualize them, and label them. c) Thinking of the Ampere-Maxwell law, in order to find B(R) you would be wise to choose a circle at radius R+ A+ ε and take the limit as ε O. Having done that, there are an infinite number of imaginary surfaces with the same closed path as their boundary. Three simple examples would be a plane through the equator, a hemisphere going just above the sphere (shown by the dashed line), or one just below it (dotted line). First consider the plane surface: i) Choose a direction for its area vector, and draw and label it. (1) Then what is the positive direction around the circle - CW or CCW from above? ii) Is there any current going through this plane? If so how much? (Think carefully about where the charge is going!) ii) What quantity do you need to calculate in order to use the second part of the Ampere-Maxwell law? What is its value? Is it zero or not? Think carefully about the relative vector directions. iv) Find the magnitude of B. (1) Is it constant or changing? v) What will be the direction of the magnetic field just outside the equator? Draw vectors and/or circled dots or crosses in a few places to show its direction, and label them. d) Next consider the hemisphere below the sphere, using the same general direction for da vectors. i) What are the contributions of the first and second terms of the Ampere-Maxwell law? ii) Find B. How does it compare to the answer above? e) Finally consider the hemisphere above the sphere, using the same general direction for dá vectors. i) What are the contributions of the first and second terms of the Ampere-Maxwell law? ii) Find B. How does it compare to the answers above?

Answer 1

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