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Chapter 26: Problem 17

17\. X-Ray Tube A certain x-ray tube operates at a current of \(7.0\) \(\mathrm{mA}\) and a potential difference of \(80 \mathrm{kV}\). What is its power in watts?

Short Answer

Expert verified
The power of the x-ray tube is 560 W.

Step by step solution

01

- Identify Given Values

The problem gives two values: the current (\(I\)) which is 7.0 mA, and the potential difference (\(V\)) which is 80 kV. Convert these values to standard units. Current: 7.0 mA = 7.0 x 10^{-3} A. Potential difference: 80 kV = 80 x 10^3 V.
02

- Recall the Power Formula

The power (\(P\)) in an electrical circuit can be calculated using the formula: \[ P = V \times I \]
03

- Substitute Values

Substitute the given values into the power formula. \[ P = (80 \times 10^3 \, V) \times (7.0 \times 10^{-3} \, A) \]
04

- Calculate the Power

Calculate the multiplication: \[ P = 80 \times 10^3 \times 7.0 \times 10^{-3} = 560 \, W \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Current and Potential Difference
To understand the power calculation of an x-ray tube, you need to know about current and potential difference.
Current, measured in Amperes (A), represents the flow of electric charge. In this problem, the current (\(I\)) is given as 7.0 mA, which means milli-Amperes.
Potential difference, also known as voltage, indicates the difference in electric potential between two points. It is measured in Volts (V). In this exercise, the potential difference (\(V\)) is 80 kV, which stands for kilo-Volts.
Hence, the values given are:
  • Current (\(I\)) = 7.0 mA
  • Potential difference (\(V\)) = 80 kV
Both these quantities need to be converted to standard units before proceeding further.
Power Formula in Electrical Circuits
The power (\(P\)) in electrical circuits can be calculated using a simple formula:
\[ P = V \times I \ \text{where } P \text{ is power, V is voltage, and I is current} \ \ \ \ \]
This relationship helps us determine how much power is being utilized by the x-ray tube. Using the given values of current and potential difference, the calculation becomes straightforward when we substitute the values into the formula.
However, remember to first convert the given values to standard units:
  • Current (\(I\)) = 7.0 mA 脳 10^{-3} = 0.007 A
  • Potential difference (\(V\)) = 80 kV 脳 10^3 = 80,000 V
Hence, to find the power:
\[ P = (80,000 \text{ V}) \times (0.007 \text{ A}) \ = 560 \text{ W} \ \ \ \ \ \ \ \ \ \ \ \]
Unit Conversion in Physics
Unit conversion is a crucial step in problems involving measurements of different magnitudes.
In the context of electrical circuits, it's essential to convert all quantities to their standard units (SI units) before performing any calculation. For instance:
  • To convert milli-Amperes (mA) to Amperes (A), multiply by 10^{-3}.
  • To convert kilo-Volts (kV) to Volts (V), multiply by 10^3.
Let鈥檚 see how it applies here:
Current (\(I\)) given: 7.0 mA becomes 7.0 脳 10^{-3} = 0.007 A.
Potential difference (\(V\)) given: 80 kV becomes 80 脳 10^3 = 80,000 V.
With these conversions made, you can confidently substitute the values into the power formula to find the output in watts.
Always take care of unit conversion first; else, the result will be erroneous and misleading.

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Most popular questions from this chapter

20\. Thermal Energy Thermal energy is produced in a resistor at a rate of \(100 \mathrm{~W}\) when the current is \(3.00 \mathrm{~A}\) What is the resistance?

24\. Heating Element A heating element is made by maintaining a potential difference of \(75.0 \mathrm{~V}\) across the length of a Nichrome wire that has a \(2.60 \times 10^{-6} \mathrm{~m}^{2}\) cross section. Nichrome has a resistivity of \(5.00 \times 10^{-7} \Omega \cdot \mathrm{m} .\) (a) If the element dissipates \(5000 \mathrm{~W}\), what is its length? (b) If a potential difference of \(100 \mathrm{~V}\) is used to obtain the same dissipation rate, what should the length be?

26\. \(100 \mathrm{~W}\) Lightbulb A \(100 \mathrm{~W}\) lightbulb is plugged into a standard \(120 \mathrm{~V}\) outlet. (a) How much does it cost per month to leave the light turned on continuously? Assume electric energy costs \(12 \mathrm{~d} / \mathrm{kW} \cdot \mathrm{h} .\) (b) What is the resistance of the bulb? (c) What is the current in the bulb? (d) Is the resistance different when the bulb is turned off?

30\. Small But Measurable A small but measurable current of \(1.2 \times 10^{-10} \mathrm{~A}\) exists in a copper wire whose diameter is \(2.5 \mathrm{~mm}\). Assuming the current is uniform, calculate (a) the current density and (b) the average electron speed.

6\. Trolley Car A steel trolley-car rail has a cross-sectional area of \(56.0 \mathrm{~cm}^{2} .\) What is the resistance of \(10.0 \mathrm{~km}\) of rail? The resistivity of the steel is \(3.00 \times 10^{-7} \Omega \cdot \mathrm{m}\).
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