Understanding the Current Equation

Oct 13, 2025 by miodeya.com 73 views

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Introduction

In this article, we will explore the concept of power delivered to an element in an electrical circuit. We will use the given current and voltage equations to calculate the power at a specific time. The current entering the positive terminal of the element is given by the equation $i = 5 \cos 60 \pi t \text{ A}$ , and the voltage $V$ is related to the current by the equation $V = 2i$ . Our goal is to find the power delivered to the element at $t = 5 \text{ ms}$ .

Understanding the Current Equation

The current equation is given by $i = 5 \cos 60 \pi t \text{ A}$ . This is a sinusoidal function, which means that the current varies sinusoidally with time. The amplitude of the current is $5 \text{ A}$ , and the angular frequency is $60 \pi \text{ rad/s}$ . The time period of the current is given by $T = \frac{2\pi}{\omega} = \frac{2\pi}{60\pi} = \frac{1}{30} \text{ s}$ .

Understanding the Voltage Equation

The voltage equation is given by $V = 2i$ . This means that the voltage is directly proportional to the current. Since the current is a sinusoidal function, the voltage will also be a sinusoidal function.

Calculating the Power

The power delivered to the element is given by the equation $P = VI$ . We can substitute the given current and voltage equations into this equation to get:

P = (2i)i = 2i^2

Substituting the current equation, we get:

P = 2(5 \cos 60 \pi t)^2 = 50 \cos^2 60 \pi t

Evaluating the Power at $t = 5 \text{ ms}$

We want to find the power delivered to the element at $t = 5 \text{ ms}$ . We can substitute this value into the power equation:

P = 50 \cos^2 60 \pi (5 \times 10^{-3})

First, we need to evaluate the argument of the cosine function:

60 \pi (5 \times 10^{-3}) = 300 \pi \times 10^{-3} = 0.9425 \text{ rad}

Now, we can evaluate the cosine function:

\cos 0.9425 \approx 0.342

Finally, we can evaluate the power:

P = 50 (0.342)^2 \approx 5.85 \text{ W}

Conclusion

In this article, we used the given current and voltage equations to calculate the power delivered to an element at a specific time. We found that the power delivered to the element at $t = 5 \text{ ms}$ is approximately $5.85 \text{ W}$ .

References

[1] "Electrical Circuits" by James W. Nilsson and Susan A. Riedel
[2] "Electric Circuits" by James M. Fiore

Additional Resources

[1] "Power in AC Circuits" by HyperPhysics
[2] "AC Circuits" by The Physics Classroom

Discussion

What do you think about the concept of power delivered to an element in an electrical circuit? Have you ever worked with AC circuits before? Share your thoughts and experiences in the comments below!

Introduction

In our previous article, we explored the concept of power delivered to an element in an electrical circuit. We used the given current and voltage equations to calculate the power at a specific time. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the difference between power and energy in an electrical circuit?

A: Power and energy are related but distinct concepts in an electrical circuit. Power is the rate at which energy is transferred or converted in a circuit, typically measured in watts (W). Energy, on the other hand, is the total amount of work done in a circuit, typically measured in joules (J). For example, if a circuit has a power of 100 W, it means that 100 joules of energy are being transferred or converted per second.

Q: How do you calculate the power delivered to an element in an AC circuit?

A: To calculate the power delivered to an element in an AC circuit, you need to know the current and voltage equations. The power equation is given by P = VI, where P is the power, V is the voltage, and I is the current. You can substitute the given current and voltage equations into this equation to get the power equation.

Q: What is the significance of the power factor in an AC circuit?

A: The power factor is the ratio of the real power (P) to the apparent power (S) in an AC circuit. It is a measure of how effectively the current is being used to perform work in the circuit. A power factor of 1 means that the current is being used 100% effectively, while a power factor of 0 means that the current is not being used at all.

Q: How do you calculate the power factor in an AC circuit?

A: To calculate the power factor in an AC circuit, you need to know the real power (P) and the apparent power (S). The power factor is given by the equation PF = P/S. You can calculate the real power using the power equation P = VI, and the apparent power using the equation S = V/I.

Q: What is the difference between a series circuit and a parallel circuit?

A: A series circuit is a circuit in which the components are connected one after the other, so that the current flows through each component in sequence. A parallel circuit, on the other hand, is a circuit in which the components are connected between the same two points, so that the current flows through each component simultaneously.

Q: How do you calculate the power delivered to an element in a series circuit?

A: To calculate the power delivered to an element in a series circuit, you need to know the current and voltage equations. The power equation is given by P = VI, where P is the power, V is the voltage, and I is the current. You can substitute the given current and voltage equations into this equation to get the power equation.

Q: How do you calculate the power delivered to an element in a parallel circuit?

A: To calculate the power delivered to an element in a parallel circuit, you need to know the current and voltage equations. The power equation is given by P = VI, where P is the power, V is the voltage, and I is the current. You can substitute the given current and voltage equations into this equation to get the power equation.

Q: What is the significance of the power triangle in an AC circuit?

A: The power triangle is a graphical representation of the power in an AC circuit. It shows the relationship between the real power (P), the reactive power (Q), and the apparent power (S). The power triangle is a useful tool for analyzing and designing AC circuits.

Q: How do you calculate the power triangle in an AC circuit?

A: To calculate the power triangle in an AC circuit, you need to know the real power (P), the reactive power (Q), and the apparent power (S). The power triangle is given by the equation P^2 + Q^2 = S^2. You can calculate the real power using the power equation P = VI, the reactive power using the equation Q = VIsin(θ), and the apparent power using the equation S = V/I.