Notes on Electricity and Magnetism I - Matrices, Vectors, Variances | PHY 481, Study notes of Physics

Download Notes on Electricity and Magnetism I - Matrices, Vectors, Variances | PHY 481 and more Study notes Physics in PDF only on Docsity! PHY481 - Lecture 2 Sections 2.1 and 2.2 of Pollack and Stump (PS) A. Co-ordinate systems we will use We shall be using three orthogonal co-ordinate systems, cartesion, cylindrical and spherical polar. We need the transformations between these systems. We have, 1. Cartesian co-ordinates ~x = (x, y, z) = x̂i + yĵ + zk̂ (1) 2. Cylindrical co-ordinates ~x = (r, θ, z) = rr̂ + zk̂ (2) x = rcosθ; y = rsinθ; x2 + y2 = r2 (3) 3. Spherical polar co-ordinates ~x = (r, θ, φ) = rr̂ (4) x = rcosφsinθ; y = rsinφsinθ; z = rcosθ; x2 + y2 + z2 = r2 (5) Following convention and as used in PS r and θ have different meanings for cylindrical as opposed to the spherical polar case. B. Rotation matrices Rotation matrices are used to rotate a co-ordinate system about an axis. We may rotate about the x, y, or z axes and there is a different matrix for each case. However the matrices are very similar so we only need to consider one in detail. Lets consider rotation about the z axis. Consider that in the original co-ordinate system the unit vectors are î, ĵ, hatk. We then rotate the co-ordinate system through an angle θ about the z-axis. In this new rotated(primed) co-ordinate system, the new unit vectors along the x′, y′, z′ directions are î′, ĵ′, k̂′. A vector ~x may be written in either of these co-ordinate systems, ie, ~x = x̂i + yĵ + zk̂ = x′î′ + y′ĵ′ + z′k̂′ (6) 1 Now it is easy to show that the relationships between the unit vectors in the original and rotated co-ordinate systems are, î′ = cosθî + sinθĵ; ĵ′ = −sinθî + cosθĵ; k̂′ = k̂ (7) Substituting these expressions in the last of Eq. (6), we find that, ~x = (x′cosθ − y′sinθ)̂i + (x′sinθ + y′cosθ)ĵ + z′k̂ = x̂i + yĵ + zk̂ (8) so that, x = x′cosθ − y′sinθ; y = x′sinθ + y′cosθ; z = z′ (9) or x′ = xcosθ + ysinθ; y′ = −xsinθ + ycosθ; z′ = z (10) The latter equation may be written in matrix form,       x′ y′ z′       =       cosθ sinθ 0 −sinθ cosθ 0 0 0 1             x y z       or ~x′ = R~x (11) where R is the rotation matrix in the middle equation. It is easy to show that the inverse of R is the transpose of R, so that, R T R = I (12) where I is the identity matrix. Rotation matrices for co-ordinate rotations by angle θ around the x,y,z axis are respectively,       1 0 0 0 cosθ sinθ 0 −sinθ cosθ             cosθ 0 −sinθ 0 1 0 sinθ 0 cosθ             cosθ sinθ 0 −sinθ cosθ 0 0 0 1       (13) C. What is a vector? What is an invariant? A vector is a quantity that behaves like the position vector ~x under co-ordinate rotations. An invariant is unaltered under rotations of the co-ordinate system. For example, we expect the dot product of two vectors and the cross product of two vectors to be unaltered by rotations of the co-ordinate system as they depend on the angle between the two vectors and not on the angle itself - they should be invariant. E.g. to check 2