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1 Chapter 22--Examples

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2 Problem A charge, +q, is surrounded by a thin, spherical shell of radius, a, which has a charge density of – on its surface. This shell is, in turn, surrounded by another thin shell of radius, b, which has a surface charge of + . Find the electric fields in Region 1: r b +q a b -- ++

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3 Step 1: Pick your shape I choose spherical! So

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4 Region 1: r
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5 Region 2: a<=r<=b q enclosed =q+(4 a 2 )*(- ) +q a b -- ++

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6 Region 3: r>b q enclosed =q+(4 a 2 )*(- )+ (4 b 2 )*( ) +q a b -- ++

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7 Problem An electric filed given by E=4i-3(y 2 +2)j pierces the Gaussian cube shown below. (E is in newtons/coulomb and y is meters). What net charge is enclosed by the Gaussian cube? X=1.0 m X=3.0 m x y z

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8 First, let’s get a sense of direction Planes 1. y-z: Normal to +x 2. x-z: Normal to –y 3. y-z: Normal to –x 4. x-z: Normal to +y 5. x-y: Normal to +z 6. x-y: Normal to -z X=1.0 m X=3.0 m x y z 1 2 5 3 4 6

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9 Need to find q enclosed

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10 Integrating each side (start with surface 1) Region 3, in which the normal vector points in the opposite direction, will have a value of -16

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11 The rest of the sides Since E is perpendicular to sides 5 & 6, the result is zero.

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12 Problem The figure below shows a cross- section of two thin concentric cylinders with radii of a and b where b>a. The cylinders equal and opposite charges per unit length of. a) Prove that E = 0 for r>a b) Prove that E=0 for r>b c) Prove that, for a

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13 First, I choose a shape I choose cylindrical! So

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14 For r
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15 For a

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16 For r>b q enclosed = L- L=0 a b This is the principle of a coaxial cable

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17 Problem A very long, solid insulating cylinder with radius R has a cylindrical hole with radius, a, bored along its entire length. The axis of the hole is a distance b from the axis of the cylinder, where a**
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**

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18 First, let’s do a solid cylinder of radius, R

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19 Now what if we have an off-axis cylinder We learned in Phys 250, that we can “translate” coordinates by r’=r-b Where b is the direction and distance of the center of the off-axis cylinder r is a vector from the origin b r r’

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20 E hole = E solid cylinder -E off-axis hole All of these are constants and do not depend on r.

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