Question:
When an AC source of emf $e =E_0$ sin (100t) is connected
across a circuit, the phase difference between the emf e and
the current i in the circuit is observed to be $ \frac{\pi}{4}$ ahead, as
shown in the diagram. If the circuit consists possibly only of
R-C or R-L or L-C in series, find the relationship between the
two elements.
When an AC source of emf $e =E_0$ sin (100t) is connected
across a circuit, the phase difference between the emf e and
the current i in the circuit is observed to be $ \frac{\pi}{4}$ ahead, as
shown in the diagram. If the circuit consists possibly only of
R-C or R-L or L-C in series, find the relationship between the
two elements.
Updated On: Oct 25, 2023
- $R= 1 k \Omega, \, C = 10 \mu F$
- $R= 1 k \Omega, \, C = 1 \mu F$
- $R= 1 k \Omega, \, L = 10 H$
- $R= 1 k \Omega, \, L = 1 H$
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The Correct Option is A
Solution and Explanation
As the current i leads the emf e by $ \frac{\pi}{4}$, it is an R-C circuit
$tan \phi = \frac{X_c}{R} \, \, or \, \, tan \frac{\pi}{4} = \frac{\frac{1}{? C}}{R}$
$\therefore \, \, \, \, \, \, \, \, \, \, \, \, \, ? CR = 1$
$As \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, ? 100 rad/s$
The product of C-R should be $\frac{1}{100}s^{-1}$
$\therefore$ Correct answer is (a).
$tan \phi = \frac{X_c}{R} \, \, or \, \, tan \frac{\pi}{4} = \frac{\frac{1}{? C}}{R}$
$\therefore \, \, \, \, \, \, \, \, \, \, \, \, \, ? CR = 1$
$As \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, ? 100 rad/s$
The product of C-R should be $\frac{1}{100}s^{-1}$
$\therefore$ Correct answer is (a).
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Concepts Used:
Phasor Diagrams
It is a representation of an oscillating quantity. In phase space with an angular velocity equal to the angular frequency of the original trigonometric function.
The projection of the phasor onto an axis at a specific time gives the value of the quantity at that time. Phasor diagrams are used in simple harmonic motion and RLC circuits which have elements that are out of phase with one another and thus difficult to work with in configuration space.
Read More: Phasor Representation
Three ways to represent phasors:
- Polar Form: Assume we have a phasor with an amplitude of Vm and a polar form that makes an angle with the horizontal axis. In the polar form we can write it as Vm.
- Rectangular Form: Any phasor can be shown as a complex number, such as A+iB.
- Exponential Form: It is represented as VmejΦ