Problem 75 - Fundamentals of Physics

Question

A certain \(x\) -ray tube operates at a current of \(7.00 \mathrm{~mA}\) and a potential difference of \(80.0 \mathrm{kV}\). What is its power in watts?

Step-by-Step Solution

Answer

The power of the \(x\)-ray tube is 560 watts.

Step 1: Understand the Given Data

We are given the current, which is the flow of electric charge and is measured in milliampere (mA), and the potential difference (or voltage), which is provided in kilovolts (kV). Specifically, the current (\(I\)) is \(7.00 \, \mathrm{mA}\) and the voltage (\(V\)) is \(80.0 \, \mathrm{kV}\). Our task is to determine the power consumed by the \(x\)-ray tube.

Step 2: Convert Units to Standard Units

For calculations in power, we need to convert current from milliamperes to amperes and voltage from kilovolts to volts. \(1 \, \mathrm{mA} = 0.001 \, \mathrm{A}\) and \(1 \, \mathrm{kV} = 1000 \, \mathrm{V}\). Therefore, the current \(I = 7.00 \, \mathrm{mA} = 7.00 \times 0.001 \, \mathrm{A} = 0.007 \, \mathrm{A}\). The voltage \(V = 80.0 \, \mathrm{kV} = 80.0 \times 1000 \, \mathrm{V} = 80000 \, \mathrm{V}\).

Step 3: Apply the Power Formula

Power (\(P\)) in an electrical circuit is calculated using the formula: \[ P = V \times I \] where \(V\) is the potential difference in volts, and \(I\) is the current in amperes.

Step 4: Calculate the Power

Substitute the converted values of current and voltage into the power formula: \[ P = 80000 \, \mathrm{V} \times 0.007 \, \mathrm{A} = 560 \, \mathrm{W}\] This result shows that the power consumed by the \(x\)-ray tube is 560 watts.

Key Concepts

x-ray tube

An X-ray tube is a device that generates X-rays by accelerating electrons from a cathode to a metal anode, typically using a high potential difference. When these electrons strike the anode, X-rays are produced due to the sudden deceleration of the electrons. This process is fundamental in medical imaging and various industrial applications. To operate efficiently, the X-ray tube requires a specific electric power, which is determined by the electrical current and the potential difference applied across the tube.

electric current

Electric current is the flow of electric charge through a conductor, such as the wire in an X-ray tube. It is measured in amperes (A), but sometimes smaller units like milliamperes (mA) are used, especially in precision instruments. To convert milliamperes to amperes, remember that:

• 1 milliampere (mA) is equal to 0.001 amperes (A).

Thus, an X-ray tube operating at 7.00 mA would have a current of 0.007 A. Understanding electric current is crucial for determining how much power the X-ray tube consumes during operation.

potential difference

Potential difference, also known as voltage, is the work done to move a charge between two points in an electric field. It is measured in volts (V). In the context of an X-ray tube, a high potential difference is required to provide the necessary energy to accelerate electrons from the cathode to the anode. Typical units of measurement include kilovolts (kV), with 1 kV equal to 1,000 volts (V).

• In our problem, the X-ray tube operates at 80.0 kV, which equals 80,000 V when converted to volts.

This high voltage is essential for producing the high-energy X-rays needed for penetrating materials.

unit conversion

Unit conversion is a critical skill in physics and engineering, allowing different units of measurement to be translated into standard base units. When calculating electric power, it is essential to convert given units like milliamperes and kilovolts into the base units of amperes and volts.

• Convert milliampere (mA) to ampere (A) using 1 mA = 0.001 A.
• Convert kilovolt (kV) to volt (V) using 1 kV = 1,000 V.

For example, in the exercise problem, a conversion was necessary for the current (7.00 mA to 0.007 A) and voltage (80.0 kV to 80,000 V). Proper unit conversion ensures accurate application of formulas, like calculating electric power using the formula \( P = V \times I \).