The Physics
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Opus in profectus

RL Circuits

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Discussion

energizing

deenergizing

Start with Kirchhoff's circuit rule.

V = IR + L  dI
dt
0 = IR + L  dI
dt

Separate the variables.

V − IR  =  dI  
L dt  
 
V/R − I  =  dI  
L/R dt  
 
−  dt  =  dI
L/R I − V/R
−  IR  =  dI
L dt
 
−  I  =  dI
L/R dt
 
−  dt  =  dI
L/R I

Integrate both sides over the appropriate limits.

  t     I    


dt  = 

dI  
L/R I − V/R
  0     0    
−  t  = ln

I − V/R

L/R − V/R
  t     I    


dt  = 

dI  
L/R I
  0     I0    
−  t  = ln

I

L/R I0

Make both sides into a power ofe.

eRt/L = 1 −  IR
V
eRt/L =  I
I0

Solve for current as a function of time.

I =  V  

1 − eRt/L

R
I = I0eRt/L  
 

The end.