The root-mean-square current Irms in the RC circuit is given by
Irms = Vrms/√R2 +X2C…… (1)
Here, Vrms is the root-mean-square voltage, R is the resistance and XC is the capacitive reactance.
The capacitive reactance XC is given by XC = 1/ωC …… (2)
Here, ω is the angular frequency and C is the capacitance.
We have given that the initial angular frequency of the source is ω and the current is I .
Substitute I for Irms and 1/ωC for XCin equation (1).
I = Vrms√R2+(1ω/C)2
⇒ I = Vrms/√R2+1ω2C2
⇒ I = VrmsωC/√R2ω2C2+1 …… (3)
When the angular frequency of the source is changed to ω/3 then the current becomes I/2 but the potential is the same.
Hence, the above equation becomes
⇒ I/2 = Vrmsω/3C/√R2(ω/3)2C2+1
⇒ I/2 = VrmsωC/√R2ω2C2 + 9 …… (4)
Divide equation (3) by equation (4). ⇒
I/I/2 = VrmsωC/√R2ω2C2 + 1/VrmsωC/√R2ω2C2 + 9
⇒2 = √R2ω2C2 + 9 / √R2ω2C2 + 1
Take square on both sides of the above equation.
⇒ 4=R2ω2C2+9 / R2ω2C2 + 1
⇒ 4R2ω2C2+4 = R2ω2C2+9
⇒ 4R2ω2C2−R2ω2C2=9−4 ⇒3R2ω2C2=5
⇒ R2ω2C2 = 5/3
⇒ R2=5/3 1/ω2C2
Take square root on both sides of the above equation.
⇒R =√5/3 1/ωC
Substitute XC for 1/ωC in the above equation.
XC/R = √3/5
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