hw4 - Problem 7. Four identical charges q, initially widely...

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Problem 7. Four identical charges q , initially widely separated, are brought to the vertices of a tetrahedron of side a (Fig. 26-26). Find the electrostatic energy of this configuration. FIGURE 26-26 Problem 7. Solution There are six different pairs of equal charges and the separation of any pair is a . Thus, W = pairs kq i q j = a = 6 2 = a . (See Problem 1.) Problem 8. A charge Q 0 is at the origin. A second charge, Q x = 2 Q 0 , is brought to the point x = a , y = 0. Then a third charge Q y is brought to the point x = 0, y = a . If it takes twice as much work to bring in Q y as it did Q x , what is Q y in terms of Q 0 ? Solution The work necessary to bring up Q x is W x = kQ 0 Q x = a = 2 kQ 0 2 = a , while the work necessary to subsequently bring up Q y is W y = 0 Q y = a + x Q y = a = 0 Q y (1 + ) = a . If W y = 2 W x , then Q y + ) = 4 Q 0 , or Q y = 4 Q 0 = + 1) = 1.66 Q 0 . (Note: 1 = + = 1.) Problem 10. Two square conducting plates measure 5.0 cm on a side. The plates are parallel, spaced 1.2 mm apart, and initially uncharged. (a) How much work is required to transfer 7.2 μ C from one plate to the other? (b) How much work is required to transfer a second 7.2 C? Solution The separation is much smaller than the linear dimensions of the plates, so the discussion in Section 26-2 applies. (a) From Equation 26-2, W = Q 2 d = 2 ε 0 A = (7.2 C) 2 (1.2 mm) = 2(8.85 × 10 12 F/m)(5 cm) 2 = 1.41 J. (b) The additional work required to double the charge on each plate is Δ W = ( 2 Q ) 2 d = 2 0 A W = 3 W = 4 . 22 J. Problem 13. A conducting sphere of radius a is surrounded by a concentric spherical shell of radius b . Both are initially uncharged. How much work does it take to transfer charge from one to the other until they carry charges ± Q ?
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Solution When a charge q (assumed positive) is on the inner sphere, the potential difference between the spheres is V = kq ( a 1 b 1 ). (See the solution to Problem 25-63(a).) To transfer an additional charge dq from the outer sphere requires work dW = V dq , so the total work required to transfer charge Q (leaving the spheres oppositely charged) is W = 0 Q V = 0 Q ( a 1 b 1 ) = 1 2 kQ 2 ( a 1 b 1 ). (Incidentally, this shows that the capacitance of this spherical capacitor is 1 = k ( a 1 b 1 ) = ab = k ( b a ) ; see Equation 26- 8a.) Problem 15. Two conducting spheres of radius a are separated by a distance l À a ; since the distance is large, neither sphere affects the other’s electric field significantly, and the fields remain spherically symmetric. (a) If the spheres carry equal but opposite charges ± q , show that the potential difference between them is 2 = a . (b) Write an expression for the work dW involved in moving an infinitesimal charge dq from the negative to the positive sphere. (c) Integrate your expression to find the work involved in transferring a charge Q from one sphere to the other, assuming both are initially uncharged.
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hw4 - Problem 7. Four identical charges q, initially widely...

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