In a circuit consisting of a capacitance and a generator with alternating emf Eg = Eg0 sin(wt), VC a

Q: What does the angle $\omega t$ represent?

It represents the instantaneous phase of the alternating voltage. As time $t$ increases, the phasor rotates counterclockwise with an angular velocity $\omega$.

Q: Why is the angle between $V_C$ and $I_C$ exactly $90^\circ$?

Mathematically, $I_C = C \frac{dV_C}{dt}$. If $V_C$ is a sine function, its derivative is a cosine function. The cosine function is exactly $90^\circ$ ahead of the sine function.

Q: How do I know which direction is 'leading' on a phasor diagram?

Phasors rotate counterclockwise. Therefore, a phasor that is further counterclockwise is 'leading' the other.

Q: Would this diagram change if the circuit had some resistance?

Yes! If there were resistance, the circuit would not be purely capacitive. The phase angle by which current leads voltage would be less than $90^\circ$.

Question

In a circuit consisting of a capacitance and a generator with alternating emf $E_g = E_{g_0} \sin\omega t$, $V_C$ and $I_C$ are the voltage and current. Correct phasor diagram for such circuit is

Sigma Percentile · Accepted Answer

### The Dance of Phasors

Imagine you are watching a radar screen where two blips are constantly rotating in a circle. These rotating vectors are what we call **phasors**, and they are the ultimate tool for understanding Alternating Current (AC) circuits.

In an AC circuit, the voltage and current are constantly changing direction and magnitude. Instead of dealing with messy sine and cosine waves on a graph, we represent them as arrows (phasors) rotating counterclockwise at an angular frequency $\omega$. The angle a phasor makes with the positive x-axis at any instant $t$ is its **phase**, given by $\omega t$.

### The Purely Capacitive Circuit

In our problem, we have a circuit with only a capacitor and an AC generator. The generator provides an alternating electromotive force (emf) given by:
$$E_g = E_{g_0} \sin(\omega t)$$

This tells us that the voltage across the capacitor, $V_C$, is perfectly in sync with the generator. Its phasor will be at an angle of $\omega t$ with the x-axis.

But what about the current, $I_C$?

Think of a capacitor as a water tank. To build up water pressure (which represents voltage), water must flow into the tank first (which represents current). Therefore, the flow of current must *precede* the buildup of voltage.

Mathematically, the current through a capacitor is the rate of change of voltage:
$$I_C = C \frac{dV_C}{dt}$$

If we take the derivative of our sine-wave voltage, we get a cosine wave:
$$I_C = C E_{g_0} \omega \cos(\omega t)$$

Using trigonometry, we know that $\cos(\omega t) = \sin(\omega t + \frac{\pi}{2})$. This $+ \frac{\pi}{2}$ is the magic key! It tells us that the current **leads** the voltage by exactly $90^\circ$ (or $\frac{\pi}{2}$ radians).

### Constructing the Final Diagram

Now, let's put it all together on our radar screen:
1. We draw the voltage phasor $V_C$ at an angle $\omega t$ in the first quadrant.
2. Since the current leads the voltage, we must draw the current phasor $I_C$ exactly $90^\circ$ *counterclockwise* (ahead) of $V_C$.

This places $I_C$ in the second quadrant, making an angle of $\omega t + \frac{\pi}{2}$ with the positive x-axis. Looking at the given options, this perfectly matches the phasor diagram shown in option (c).

Always remember this golden rule for capacitors: **ICE** (Current $I$ leads Capacitive voltage $E$). It will save you every time!

Animated Solution for Physics - Alternating Current: In a circuit consisting of a capacitance and a generator with alternating emf $E_{g} = E_{g_{0}} sin ω t$ , $V_{C}$ and $I_{C}$ are the voltage and current. Correct phasor diagram for such circuit is

Select Answer:

Visualized Solution

Equation of Alternating EMF

Phase Relationship in a Purely Capacitive Circuit

Constructing the Phasor Diagram

The Sigma Insight: AC Circuits and Power in AC Circuits

The Dance of Phasors

The Purely Capacitive Circuit

Constructing the Final Diagram

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