Application of High Magnetic Fields in Semiconductor Physics

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Since the current density times when multiplied by the cross-sectional area gives the total current flowing through the element, we can write A¼ m0 dl I m dl I or A ¼ 0 4p r 4p r ð1:137Þ The above equation gives the vector potential at a given point due to a current element of length dl. 9 Force on a charged particle Consider a region where there is both an E- and a B-fields. Assume that at any given instant a charged particle of charge q is at point (x, y, z) and it is moving with velocity v. The particle will experience a force both due to E- and B-fields.

Consider the operation: rSðx; y; zÞ ¼ ax @ @ @ Sðx; y; zÞ þ ay Sðx; y; zÞ þ az Sðx; y; zÞ @x @y @z ð1:21Þ Observe that when the nabla operator operates on a scalar function it gives rise to a vector. The resulting vector is called the gradient of the scalar function S(x, y, z) and is denoted by Grad S. We can also define the same scalar function either in cylindrical or in spherical coordinate systems. 2 @ 1 @ 1 @ Sðr; q; jÞ þ ay Sðr; q; jÞ þ aj Sðr; q; jÞ @r r @q rsin q @j ð1:23Þ The divergence of a vector field The scalar product of the nabla vector with a vector field F is defined as the divergence of that vector field.

10. Consider a current element dl. e. dl ¼ jdlj. The direction of the vector dl is in the direction of current flow. 10 The geometry necessary to calculate the vector potential at point P due to a current element of length dl and of cross-section da flowing uniformly across the cross-section of the element. The current density in the element is given by J. The vector potential due to the current element at a distance r when r ) dl is A¼ m0 J ðdl Á daÞ 4p r ð1:136Þ where da is the cross-sectional area of the element.

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