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19 Force and Torque 1

# 19 Force and Torque 1 - Lecture 21 Force and Torque Given a...

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Lecture 21 Force and Torque Given a system of initially coupled coils with a moving part ( ) L x i λ = , where x is the position of the moving part, then by the first law of thermodynamics, elec mech W W W φ = ∆ + ∆ If x changes by x at constant current i then, 1 2 T W i Li φ = where ( ) ( ) L L x x L x = + ∆ . In addition the electric energy delivered by the n external sources during the change can be obtained as, T T elec W i d i λ λ = = where ( ) ( ) L x x i L x i λ = + ∆ . Then 1 2 T mech elec W W W i Li φ = ∆ − ∆ = Let us consider an infinitesimal change x dx , and then the differential of the energy would be mech elec dW dW dW φ = If additionally we consider a differential of the current (no longer constant) then the differential of the stored energy in the field is ( ) ( ) ( ) ( ) 1 1 2 2 T T dW i di L x dx i di i L x i φ = + + + ( ) ( ) ( ) ( ) ( ) 1 1 2 2 T T T T T dW i L x dx i di L x dx i i L x dx di di L x dx di i L x i φ = + + + + + + + Since ( ) ( ) T L x dx L x dx + = + then ( ) ( ) T T di L x dx i

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19 Force and Torque 1 - Lecture 21 Force and Torque Given a...

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