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[Physics Numerical] LR Series Circuit

R l series circuit

A coil of inductance 0.50 H and resistance 100 Ω is connected to a 240 V 50 Hz ac supply.
(a) What is the maximum current in the coil?
(b) What is the time lag between the voltage maximum and the current maximum?

Given,
Inductance, L = 0.50 H
Resistance, R = 100 Ω
Effective voltage, Ev = 240 V
Frequency of the ac supply, f = 50 Hz

We can calculate,
Angular frequency, ω = 2πf = 100 π
Peak voltage, straight E sub<span><span><span>script</span></span></span> 0 space equals space square root of 2 space straight E sub<span><span><span>script</span></span></span> straight v space equals space square root of 2 space cross times 240 space straight V

(a) The maximum amount of current in the coil is given by,

straight I sub<span><span><span>script</span></span></span> 0 space equals space fraction numerator straight E sub<span><span><span>script</span></span></span> 0 over denominator square root of straight R squared plus straight omega squared straight L squared end root end fraction space space space space equals fraction numerator square root of 2 space cross times space 240 over denominator square root of 10 to the power of 4 plus left parenthesis 100 straight pi space cross times 0.5 right parenthesis squared end root end fraction space space space space equals space 1.82 space straight A


(b) In the LR circuit,

If, straight E space equals space straight E sub<span><span><span>script</span></span></span> 0 space cos space ωt space space then comma space space straight I space equals space straight I sub<span><span><span>script</span></span></span> 0 space cos space left parenthesis ωt minus straight ϕ right parenthesis
Now, At straight t space equals space 0 comma space space space straight E space equals space straight E sub<span><span><span>script</span></span></span> o space space straight i. straight e. comma space space voltage space is space maximum.
At straight t space equals space straight ϕ over straight omega comma space space straight I space equals space straight I sub<span><span><span>script</span></span></span> 0 space cos space left parenthesis straight ϕ minus straight ϕ right parenthesis space equals space straight I sub<span><span><span>script</span></span></span> 0 space cross times space 1 comma space space current space is space maximum
∴ The time lag between voltage maximum and current maximumequals space straight ϕ over straight omega
Using the below-given relation, we can find the value of ϕ.
Therefore,

box enclose tan space straight ϕ space equals space ωL over straight R end enclose space space space space space space space space space space space equals space fraction numerator 2 straight pi space cross times space 50 space cross times 0.50 over denominator 100 end fraction space space space space space space space space space space space equals space fraction numerator 22 over denominator 7 space cross times space 2 end fraction space space space space space space space space space space equals space 1.571 rightwards double arrow space space straight ϕ space equals space tan to the power of negative 1 end exponent space left parenthesis 1.571 right parenthesis space space space space space space space space space space equals space 57.5 space degree space space space space space space space space space space equals space fraction numerator 57.5 space straight pi over denominator 180 end fraction radian
Thus, 

box enclose Time space lag space equals space straight ϕ over straight omega end enclose space space space space space space space space space space space space space space space space space equals space fraction numerator 57.5 space straight pi over denominator 180 space cross times space 2 πf end fraction space space space space space space space space space space space space space space space space equals space fraction numerator 57.5 over denominator 180 space cross times 2 cross times 50 end fraction space space space space space space space space space space space space space space space space equals space space 3.19 space cross times space 10 to the power of negative 3 end exponent straight s.

is the required time lag between the maximum voltage and maximum current.