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5. The Poisson equation 8² 8² + ø(r) = - əy² əz² relates the electrostatic potential (r) = (x, y, z) to the charge distribution p(r) in vacuum. In partic- ular, when we are dealing with a point charge, we write this equation as 7²6 = 10²2 [əx² + 8² 8² 8² əz² + + 8x² p(r) G(r. r) = --—-8¹ (r-r), -1/-0 where r' = (x, y, z) is the position of the point charge, 63) (r - r') = 8(x − x')6(y-y')(z-z') is the Dirac delta function in three-dimensional space. Solve for G(r, r') by performing a three-dimensional Fourier transform on the Poisson equation, followed by a three-dimensional inverse Fourier transform. You may use the result fd = sin

5. The Poisson equation 8² 8² + ø(r) = - əy² əz² relates the electrostatic potential (r) = (x, y, z) to the charge distribution p(r) in vacuum. In partic- ular, when we are dealing with a point charge, we write this equation as 7²6 = 10²2 [əx² + 8² 8² 8² əz² + + 8x² p(r) G(r. r) = --—-8¹ (r-r), -1/-0 where r' = (x, y, z) is the position of the point charge, 63) (r - r') = 8(x − x')6(y-y')(z-z') is the Dirac delta function in three-dimensional space. Solve for G(r, r') by performing a three-dimensional Fourier transform on the Poisson equation, followed by a three-dimensional inverse Fourier transform. You may use the result fd = sin

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5. The Poisson equation 8² + ax² ay² az² relates the electrostatic potential (r) = (x, y, z) to the charge distribution p(r) in vacuum. In partic- ular, when we are dealing with a point charge, we write this equation as 7²4 = | 8² əx² + 8² 8² əz² 2² + o(r) + G 5, 8) = - = 10² G(r, r') p(r) €0 -83) (r-r'), where r' = (x, y, z) is the position of the point charge, 63) (r - r') = 6(x − x')8(y - y')6(z - z') is the Dirac delta function in three-dimensional space. Solve for G(r, r') by performing a three-dimensional Fourier transform on the Poisson equation, followed by a three-dimensional inverse Fourier transform. You may use the result 00 sin id=
Transcribed Image Text:5. The Poisson equation 8² + ax² ay² az² relates the electrostatic potential (r) = (x, y, z) to the charge distribution p(r) in vacuum. In partic- ular, when we are dealing with a point charge, we write this equation as 7²4 = | 8² əx² + 8² 8² əz² 2² + o(r) + G 5, 8) = - = 10² G(r, r') p(r) €0 -83) (r-r'), where r' = (x, y, z) is the position of the point charge, 63) (r - r') = 6(x − x')8(y - y')6(z - z') is the Dirac delta function in three-dimensional space. Solve for G(r, r') by performing a three-dimensional Fourier transform on the Poisson equation, followed by a three-dimensional inverse Fourier transform. You may use the result 00 sin id=
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