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Problem 3. Maxwell's equations for electromagnetism in free space (i.e. in the absence of charges, currents and dielectric or magnetic media) can be written ) V-B=0, i) V-E=0, - 1 9E JB m) VxE+ =0,. (iv) VxB==0. () x K+ o (1v) x =5 A vector A is defined by B=V x A, and a scalar by E =V A/t . Show that if the condition | 3 (v) VA+F$\"O 1s imposed (this 1s known as choosing the Lorentz gauge), then A and satisfy wave equations as follows. 1 8 c? 1 #A .- 2 (vnn) VA- ?F = (vi) V =0, N.B. To solve, proceed as follows. (a) Verify that the expressions for B and E 1n terms of A and are consistent with (1) and (1i1). (b) Substitute for E in (11) and use the derivative with respect to time of (v) to eliminate A from the resulting expression. Hence obtain (vi). () Substitute for B and E in (iv) in terms of A and . Then use the gradient of (v) to simplify the resulting equation and so obtain (vii)