An L-R-C circuit has an inductance of 0.430 , a capacitance of 2.45×10−5 , and a resistance of R.

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(A) What is the angular frequency of the circuit when (B) What value must R have to give a decrease in angular frequency of 6.00 compared to the value calculated in Part A?Express your answer in ohms.

Given:

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An L-R-C circuit has an inductance of 0.430 , a capacitance of 2.45×10−5 , and a resistance of R.

(A) What is the angular frequency of the circuit when (B) What value must R have to give a decrease in angular frequency of 6.00  The voltage equations for the three circuit elements are :

(1) (2)   (3)

In equation (3) , we have that is the initial voltage on the capacitor which is a continuous .

I will assume current flowing in the clockwise direction . This means we have voltage drops in the clockwise

direction . I will sum voltage rises in the counter clockwise direction . According to Kirchhoff's voltage law

the sum of these voltage rises is zero . I will start the sum at the top of the resistor : . Take the derivative of this equation with respect to , and we get : . Divide this equation through by , and wecan write the differential

equation for the series circuit as :

(4)

The auxilliary equation to solve this differential equation is :

We can solve this auxilliary equation using the quadratic formula : .

The current and voltage in the series RLC circuit oscilates when : . We can then write the auxilliary solution as : (5)

Thecoefficient of inequation (5) is the radian frequencythat we are looking for :