Problem 21 - Technical Mathematics with Calculus

Question

The current to a projection lamp is measured at 4.7 A when the line voltage is \(115.45 \mathrm{V} .\) Using (power \(=\) voltage \(\times\) current), find the power dissipated in the lamp.

Step-by-Step Solution

Answer

The power dissipated in the lamp is 542.615 watts.

Step 1: Understand the Power Formula

The power dissipated by an electrical device is calculated using the formula Power (P) = Voltage (V) x Current (I). In this formula, power is measured in watts (W), voltage in volts (V), and current in amperes (A).

Step 2: Substitute Known Values

Substitute the given values of current (I = 4.7 A) and voltage (V = 115.45 V) into the power formula.

Step 3: Calculate the Power Dissipated

Multiply the voltage by the current to find the power dissipated by the lamp. P = 115.45 V x 4.7 A.

Step 4: Perform the Multiplication

Carry out the multiplication to get the power in watts. P = 542.615 W.

Key Concepts

Power Formula

Understanding the power formula is essential when working with electrical devices. Power, denoted by the symbol \(P\), represents the rate at which energy is used or converted. It's measured in watts \(W\), named after James Watt, the Scottish inventor. The power formula, which is \(P = V \times I\), shows the relationship between power, voltage \(V\), and current \(I\).

When you're working with devices like a projection lamp, you can use this formula to determine how much energy it uses over a certain period of time. In this case, simply multiply the current flowing through the device by the voltage across it to find its power consumption. For the lamp in question, with a known current of 4.7 amperes \(A\) and voltage of 115.45 volts \(V\), the power dissipation is easily calculated. This calculation not only helps to understand the energy usage but also has practical implications, like verifying whether the lamp is within the safety ratings of an electrical circuit.

Voltage

Voltage, commonly represented by the symbol \(V\), is the difference in electric potential between two points. It can be thought of as the pressure that pushes electric charges through a conducting loop, similar to the pressure that pushes water through a pipe. Measured in volts \(V\), it's a critical component in the power formula, indicating how much potential energy exists to move electrons in an electric circuit.

A higher voltage means more potential energy and, as such, sometimes more power dissipated through a device like our projection lamp. The 115.45 volts identified in the lamp's specifications indicate a substantial amount of potential to do work, which, when combined with the current, results in the total power output. It's the voltage in an electrical circuit that can create a current flow if a complete path is given, and in the power calculation, it is always crucial to use the exact voltage value to ensure accuracy.

Current

Current, denoted by the symbol \(I\), is the flow of electric charge in a conductor. It is measured in amperes, often shortened to amps or \(A\). An ampere is defined as one coulomb of electric charge passing a given point per second.

In relation to our projection lamp, the current rating of 4.7A determines how much charge is flowing through it under normal operating conditions. Understanding the flow of electric charge is integral as it interacts with the voltage to determine the power consumption through the power formula \(P = V \times I\). The current value of 4.7A in our example, when multiplied by the voltage of 115.45V, reveals the rate at which the lamp converts electrical energy into light and heat. Accurately measuring the current is important for safe electrical installations and to ensure that electrical devices like the lamp function within intended specifications.