By James Clerk Maxwell
An unabridged, unaltered variation in seven components, to incorporate: Introductory - On Electromagnetic Induction - common Equations of the Electromagnetic box - Mechanical activities within the box - thought of Condensers - Electromagnetic conception of sunshine - Calculation of the Coefficients of Electromagnetic Induction
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Additional resources for A Dynamical Theory of the Electromagnetic Field
If the B field becomes large enough, the forward motion will be brought to zero, and the particle will be repelled out of the region of high magnetic field. 4 Motion in uniform E and B fields Having considered the case of particle motion in a uniform E field, and in a uniform B field, we may also consider the situation of motion in uniform E and B fields together. To 41 discuss this in detail is quite involved, and we shall not pursue it extensively here. e. |v| << 1. 52) and so m¨ x = eB y˙ , m¨ y = eEy − eB x˙ , m¨ z = eEz .
This accounts for the second term. Putting all this together, we have d(mγ x˙ i ) dt ∂Ai + e(∂i Aj − ∂j Ai ) x˙ j , ∂t = e(Ei + ijk x˙ j Bk ) . 66). 77), we can also instead directly derive the four-dimensional equation dpµ /dτ = eF µν Uν . 85) where Uµ is the 4-velocity. 77) gives τ2 δS = τ1 τ2 = τ1 τ2 (−mdUµ δxµ − edAµ δxµ + e∂µ Aν δxµ dxν ) , (−m = τ1 Now we have (mUµ dδxµ + eAµ dδxµ + e∂ν Aµ δxν dxµ ) , dUµ dxν dAµ −e + e∂µ Aν )δxµ dτ . 87) and so τ2 −m δS = τ1 dUµ − e∂ν Aµ U ν + e∂µ Aν U ν δxµ dτ .
15) and so the action will be S=− 1 16π Fµν F µν d4 x . 16) We can now derive the source-free Maxwell equations by requiring that this action be stationary with respect to variations of the gauge field A µ . It must be emphasised that we treat Aµ as the fundamental field here. 46 The derivation goes as follows. We shall have 1 1 (δFµν F µν + Fµν δF µν )d4 x = − , δFµν F µν d4 x , 16π 8π 1 1 F µν ∂µ δAν d4 x , − , F µν (∂µ δAν − ∂ν δAµ )d4 x = − 8π 4π 1 1 − ∂µ (F µν δAν )d4 x + (∂µ F µν ) δAν d4 x , 4π 4π 1 1 F µν δAν dΣµ + − (∂µ F µν ) δAν d4 x , 4π Σ 4π 1 (∂µ F µν ) δAν d4 x .
A Dynamical Theory of the Electromagnetic Field by James Clerk Maxwell