chap6 - CHAPTER 6 6.1 Construct a curvilinear square map...

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CHAPTER 6. 6.1 Construct a curvilinear square map for a coaxial capacitor of 3-cm inner radius and 8-cm outer radius. These dimensions are suitable for the drawing. a) Use your sketch to calculate the capacitance per meter length, assuming D R = 1: The sketch is shown below. Note that only a 9 sector was drawn, since this would then be duplicated 40 times around the circumference to complete the drawing. The capacitance is thus C . = D 0 N Q N V = D 0 40 6 = 59 pF / m b) Calculate an exact value for the capacitance per unit length: This will be C = 2 πD 0 ln ( 8 / 3 ) = 57 pF / m 84

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6.2 Construct a curvilinear-square map of the potential field about two parallel circular cylinders, each of 2.5 cm radius, separated by a center-to-center distance of 13cm. These dimensions are suitable for the actual sketch if symmetry is considered. As a check, compute the capacitance per meter both from your sketch and from the exact formula. Assume D R = 1. Symmetry allows us to plot the field lines and equipotentials over just the first quadrant, as is done in the sketch below (shown to one-half scale). The capacitance is found from the formula C = (N Q /N V )D 0 , where N Q is twice the number of squares around the perimeter of the half-circle and N V is twice the number of squares between the half-circle and the left vertical plane. The result is C = N Q N V D 0 = 32 16 D 0 = 2 D 0 = 17 . 7pF / m We check this result with that using the exact formula: C = πD 0 cosh 1 (d/ 2 a) = 0 cosh 1 ( 13 / 5 ) = 1 . 95 D 0 = 17 . 3pF / m 85
6.3. Construct a curvilinear square map of the potential field between two parallel circular cylinders, one of 4-cm radius inside one of 8-cm radius. The two axes are displaced by 2.5 cm. These dimensions are suitable for the drawing. As a check on the accuracy, compute the capacitance per meter from the sketch and from the exact expression: C = 2 πD cosh 1 ± (a 2 + b 2 D 2 )/( 2 ab) ² where a and b are the conductor radii and D is the axis separation. The drawing is shown below. Use of the exact expression above yields a capacitance value of C = 11 . 5 D 0 F / m . Use of the drawing produces: C . = 22 × 2 4 D 0 = 11 D 0 F / m 86

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6.4. A solid conducting cylinder of 4-cm radius is centered within a rectangular conducting cylinder with a 12-cm by 20-cm cross-section. a) Make a full-size sketch of one quadrant of this configuration and construct a curvilinear-square map for its interior: The result below could still be improved a little, but is nevertheless sufficient for a reasonable capacitance estimate. Note that the five-sided region in the upper right corner has been partially subdivided (dashed line) in anticipation of how it would look when the next-level subdivision is done (doubling the number of field lines and equipotentials). b) Assume D = D 0 and estimate C per meter length: In this case N Q is the number of squares around the full perimeter of the circular conductor, or four times the number of squares shown in the drawing. N V is the number of squares between the circle and the rectangle, or 5. The capacitance is estimated to be C = N Q N V D 0 = 4 × 13 5 D 0 = 10 . 4 D 0 .
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chap6 - CHAPTER 6 6.1 Construct a curvilinear square map...

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