1 Ts 1 Figure 1: A charge Q radiating an electric field E. The dashed line...

 

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1 Ts 1 Figure 1: A charge Q radiating an electric field E. The dashed line indicates a Gaussian surface in 3-D that encloses the charge. 1. [2 points] Consider a charge Q distributed over the surface of a small sphere that is centered at the origin with radius R. The electric field radiates from the surface of the sphere in all directions in 3-D space as shown in Figure 1. Using Gauss's law, calculate the electric field at a distance 7 > R from the center of the charge distribution. Hint:: this is a common example that is explicitly worked out in most physics textbooks to illustrate Gauss's law. 2. [4 points] Now, instead of radiating in 3-D space, suppose that the electric field is confined to radiate in a 2-D plane. That is, the charge Q is distributed around the perimeter of a circle of radius R and the electric field is confined to propogate in the plane of the page as shown in Figure 2. Revised: Monday 22nd February, 2016 17:30 Using Gauss's law, calculate the electric field in the plane of the page at a point > R from the center of the charge distribution Q. XA kr P TE ©2014 J. Reid Mumford ds Figure 2: A charge Q radiating an electric field Ễ in 2-D - the electric field is confined to the plane of the page. The dashed line indicates a loop P that encloses the charge. The vector ds represents a small patch of the perimeter. Qenclosed €0 Hint:: For Gauss's law in 3-D we integrate with respect to the radial vector dà to obtain the total surface area of the Gaussian surface. In 2-D, we integrate over the radial vector ds around a closed loop P to obtain the total perimeter (See Figure 2): $ ³. Ē.dÅ= (0.1) 3. [4 points] In Problem 1, you showed that The electric field for a point charge radiating in 3-dimensions has a distance dependence of 1/r². In Problem 2 you showed that the electric field for a point charge radiating in 2-dimensions has a distance dependence of 1/r. Consider again the 2-dimensional case described in Problem 2. What distance dependence do you expect for the electric potential? Justify your answer. 1 Ts 1 Figure 1: A charge Q radiating an electric field E. The dashed line indicates a Gaussian surface in 3-D that encloses the charge. 1. [2 points] Consider a charge Q distributed over the surface of a small sphere that is centered at the origin with radius R. The electric field radiates from the surface of the sphere in all directions in 3-D space as shown in Figure 1. Using Gauss's law, calculate the electric field at a distance 7 > R from the center of the charge distribution. Hint:: this is a common example that is explicitly worked out in most physics textbooks to illustrate Gauss's law. 2. [4 points] Now, instead of radiating in 3-D space, suppose that the electric field is confined to radiate in a 2-D plane. That is, the charge Q is distributed around the perimeter of a circle of radius R and the electric field is confined to propogate in the plane of the page as shown in Figure 2. Revised: Monday 22nd February, 2016 17:30 Using Gauss's law, calculate the electric field in the plane of the page at a point > R from the center of the charge distribution Q. XA kr P TE ©2014 J. Reid Mumford ds Figure 2: A charge Q radiating an electric field Ễ in 2-D - the electric field is confined to the plane of the page. The dashed line indicates a loop P that encloses the charge. The vector ds represents a small patch of the perimeter. Qenclosed €0 Hint:: For Gauss's law in 3-D we integrate with respect to the radial vector dà to obtain the total surface area of the Gaussian surface. In 2-D, we integrate over the radial vector ds around a closed loop P to obtain the total perimeter (See Figure 2): $ ³. Ē.dÅ= (0.1) 3. [4 points] In Problem 1, you showed that The electric field for a point charge radiating in 3-dimensions has a distance dependence of 1/r². In Problem 2 you showed that the electric field for a point charge radiating in 2-dimensions has a distance dependence of 1/r. Consider again the 2-dimensional case described in Problem 2. What distance dependence do you expect for the electric potential? Justify your answer.

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