Question

A lossless coaxial cable has capacitance of $ 7 \times {10^{ - 11}}F $ and an inductance of $ 0.39\mu H. $ Calculate characteristic impedance of the cable.
A. $ 65 $
B. $ 75 $
C. $ 66 $
D. $ 77 $

Answer 1

Hint: Impedance (z) is a measure of net opposition to current due to capacitance and inductance. Capacitance and inductance is given in the question. Use the relation between inductance, capacitance and impedance to solve the above question.

Formulae used:
$ Z = \sqrt {\dfrac{L}{C}} $
Where,
$ Z $ is impedance
$ L $ is inductance
$ C $ is capacitance

Complete step by step answer:
It is given in the question that,
Capacitance is, $ C = 7 \times {10^{ - 11}}F $
Inductance is, $ L = 0.39\mu H $
Since the capacitance is given in SI unit. We will convert the inductance in SI unit as well. Therefore, we get
  $ L = 0.36 \times {10^{ - 6}}H $
We know that, for a circuit with capacitor and inductor. Impedance is the square root of the ratio of inductance to capacitance.
Thus, the impedance is given by
  $ Z = \sqrt {\dfrac{L}{C}} $
Where,
  $ Z $ is Impedance
  $ L $ is Inductance
  $ C $ is capacitance
By substituting the given value in the above equation, we get
  $ Z = \sqrt {\dfrac{{0.39 \times {{10}^{ - 6}}}}{{7 \times {{10}^{ - 11}}}}} $
By simplifying, we get
  $ Z = \sqrt {0.055 \times {{10}^{ - 6 + 11}}} $ $ \left( {\because \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}} \right) $
  $ \Rightarrow Z = \sqrt {0.055 \times {{10}^5}} $
  $ = \sqrt {55 \times {{10}^2}} $
  $ \Rightarrow Z = 74.16 $ .

Therefore, from the above explanation the correct option is (B). $ 75 $ .

Additional information:
Inductance is the tendency of an electrical conductor to oppose the change in electric current flowing through it.Capacitance is the ability of a system to store a charge. It is the ratio of change in electric charge with respect to the change in potential


Note:
You can use a log table to simplify calculations. But for this question,
Since the options are not close to each other, you can solve it by approximation. You can solve it like this.
$ Z = \sqrt {\dfrac{{0.39}}{7}} \times {10^{ - 6 + 11}} $
$ = \sqrt {\dfrac{{39000}}{7}} $
$ = \sqrt {5571} $
$ {75^2} = 5625 $
$ \therefore Z $ is closet to $ 75\Omega $ .
Hence option (B) 75.