Question

A capacitor is connected to an A.C generator. The ratio of reactance and impedance of capacitor is:
(A) $1$
(B) less than $1$
(C) greater than $1$
(D) zero

Answer 1

Hint: In general, if the capacitor is connected to an A.C generator, there are three main parameters that must be known. They are impedance, resistance and reactance. By using the reactance of a capacitor and the impedance of a capacitor formula, the ratio can be determined.

Formulae Used:
The reactance of a capacitor, ${X_c} = \dfrac{1}{{2\pi fC}}$
Where, ${X_c}$ is the reactance of the capacitor, $f$ is the frequency and $C$ is the capacitance.
The impedance of a capacitor, $Z = - \dfrac{1}{{2\pi fC}}$
Where, ${Z}$ is the impedance of the capacitor, $f$ is the frequency and $C$ is the capacitance.

Complete step-by-step solution:
Reactance of a capacitor:
The current which flows opposite to the normal current flow through an A.C capacitor is called capacitive reactance. And it is inversely proportional to the supplied frequency. Capacitors store energy in their conductive plates in the form of an electrical charge.
${X_c} = \dfrac{1}{{2\pi fC}}\,................\left( 1 \right)$
Impedance of a capacitor:
The impedance of a capacitor is equal to the magnitude of the reactance of the capacitor but these two are not identical. But the impedance of a capacitance is negative because the frequency is negative for impedance.
$Z = - \dfrac{1}{{2\pi fC}}\,..............\left( 2 \right)$
To find the ratio of the reactance of a capacitor and impedance of a capacitor, then equation (1) divided by equation (2), then,
$\dfrac{{{X_c}}}{Z} = \dfrac{{\left( {\dfrac{1}{{2\pi fC}}} \right)}}{{\left( { - \dfrac{1}{{2\pi fC}}} \right)}}$
The above equation is rearranged as,
$\dfrac{{{X_c}}}{Z} = \dfrac{1}{{2\pi fC}} \times - \dfrac{{2\pi fC}}{1}$
By cancelling the same terms, then the above equation is written as,
$\dfrac{{{X_c}}}{Z} = - 1$
Thus, the above equation shows the ratio of reactance and impedance of capacitor.
The ratio value is negative, then the ratio of reactance and impedance of capacitor is less than $1$.
Hence, the option (B) is the correct answer.

Note: The resistance of an ideal capacitor is zero. The reactance of an ideal capacitor, and therefore its impedance, is negative for all frequencies and capacitance values. The effective impedance of a capacitor is dependent on the frequency, and for an ideal capacitor it always decreases with frequency.