hw1_p5405_f05 - 3 Apply the gradient operator in spherical...

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HW 1 Phys5405 F05 due Sept. 1 1) When a perfectly conducting sphere of radius R is placed in a uniform static electric field E = E 0 e z , the resulting electric field at r (r > R) is, E = E 0 [cos θ (1 + 2( R/r ) 3 ) e r - sin θ (1 - ( R/r ) 3 ) e θ ] , (1) where θ is the usual polar angle. Write the field in cartesian components and unit vectors and then in cylindrical components and unit vectors. 2) Obtain the velocity v = d r /dt = ˙r and acceleration a = d v /dt = ˙v in spherical compo- nents and unit vectors. Recall in cartesian coordinates these are, v = ˙ x e x + ˙ y e y + ˙ z e z a = ¨ x e x + ¨ y e y + ¨ z e z In problems 3-5, A,B are vector functions of coordinates.
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Unformatted text preview: 3) Apply the gradient operator in spherical coordinates, ∇ = e r ∂/∂r + e θ (1 /r ) ∂/∂θ + e φ (1 /r sin θ ) ∂/∂φ, (2) to A and obtain the divergence and curl of that vector in spherical coordinates. This is alternative to the way I did it in class. 4) Prove ∇ ( A · B ) = ( A · ∇ ) B + A × ( ∇ × B ) + ( B · ∇ ) A + B × ( ∇ × A ) 5) Prove R V dV ( ∇ × A ) = R S da ( n × A ), where S is the boundary area of volume V and n is the outward normal at surface element da. 1...
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