Problem 26 - College Physics

Question

X-ray tube. An X-ray tube is similar to a cathode-ray tube. (See previous problem.) Electrons are accelerated to high speeds at one end of the tube. If they are moving fast enough when they hit the target at the other end, they give up their energy as X-rays (a form of nonvisible light). (a) Through what potential difference should electrons be accelerated so that their speed is 1.0\(\%\) of the speed of light when they hit the target? (b) What potential difference would be needed to give protons the same kinetic energy as the electrons? (c) What speed would this potential difference give to protons? Express your answer in \(\mathrm{m} / \mathrm{s}\) and as a percent of the speed of light.

Step-by-Step Solution

Answer

(a) 255 V (b) 255 V (c) 2.2 × 10^5 m/s or 0.073% of the speed of light.

Step 1: Calculate the Speed of Electrons.

First, calculate the speed of electrons. We know the speed of light \( c = 3 \times 10^8 \, \text{m/s} \). Thus, 1.0\(\%\) of the speed of light is given by \( v = 0.01 \cdot c = 0.01 \times 3 \times 10^8 \). Therefore, \( v = 3 \times 10^6 \, \text{m/s} \).

Step 2: Relate Kinetic Energy to Potential Difference.

The kinetic energy \( KE \) of an electron can be given by \( KE = \frac{1}{2}mv^2 \), where \( m \) is the mass of an electron \( 9.11 \times 10^{-31} \text{kg} \). The potential difference \( V \) provides energy \( eV \) equal to this kinetic energy: \( eV = \frac{1}{2}mv^2 \).

Step 3: Solve for Potential Difference for Electrons.

Rearrange the equation \( eV = \frac{1}{2}mv^2 \) to solve for \( V \): \[ V = \frac{mv^2}{2e} \]. Substitute \( m = 9.11 \times 10^{-31} \text{kg} \), \( v = 3 \times 10^6 \text{m/s} \), and \( e = 1.6 \times 10^{-19} \text{C} \). Calculate to get \( V = \frac{9.11 \times 10^{-31} \times (3 \times 10^6)^2}{2 \times 1.6 \times 10^{-19}} \approx 255 \text{ V} \).

Step 4: Determine Potential Difference for Protons.

Since protons have the same kinetic energy as the electrons, the potential difference for protons \( V_p \) is the same as for electrons because \( eV_p = KE \) and \( KE \) remains unchanged. Thus, \( V_p = 255 \text{ V} \).

Step 5: Calculate Proton Speed under the Same Potential Difference.

Now find the speed of a proton provided the same \( V = 255 \text{ V} \). Use \( \frac{1}{2}mv^2 = eV \), rearrange to solve for \( v \): \[ v = \sqrt{\frac{2eV}{m_p}} \]. Substitute \( V = 255 \), \( e = 1.6 \times 10^{-19} \), and \( m_p = 1.67 \times 10^{-27} \text{kg} \). Calculate to find \( v = \sqrt{\frac{2 \times 1.6 \times 10^{-19} \times 255}{1.67 \times 10^{-27}}} \approx 2.2 \times 10^5 \text{m/s} \).

Step 6: Express Proton Speed as a Percentage of Light Speed.

The speed of light \( c = 3 \times 10^8 \text{m/s} \). Therefore, the proton's speed as a percentage is \( \frac{2.2 \times 10^5}{3 \times 10^8} \times 100 \approx 0.073\% \).

Key Concepts

Potential Difference

Understanding potential difference is crucial in the context of an X-ray tube. When electrons accelerate within the tube, they travel from a point of higher electric potential to one of lower potential.
This transition provides the energy necessary to increase the electrons' speed.

The potential difference, measured in volts (V), is the electric potential energy per unit charge. It essentially describes how much work needs to be done to move a charge within an electric field.

• Higher potential differences result in greater energy transfer to the charged particle.
• This greater energy transfer allows for higher speed and energy for the electrons when they hit the target.
• In our scenario, a potential difference of 255 V is sufficient to accelerate electrons such that they reach 1% of the speed of light.

Kinetic Energy

Kinetic energy is the energy that an object has due to its motion, and it's a key concept when discussing the movement of electrons in an X-ray tube.

For moving particles, like electrons and protons, kinetic energy (KE) is given by:\[KE = \frac{1}{2}mv^2\]where:

• \( m \) is the mass of the particle, and
• \( v \) is its velocity.

The kinetic energy can be thought of as how much work is needed to accelerate a particle to a certain speed.

In our exercise, while electrons and protons have vastly different masses, they can have the same kinetic energy if the potential difference through which they are accelerated is the same, given that potential difference is directly related to kinetic energy.

Speed of Light

In physics, especially when studying particles like electrons and protons, the speed of light, represented by \( c \), is extremely significant.
\[ c = 3 \times 10^8 \, \text{m/s} \]
This is not just the speed at which light travels in a vacuum, but also a universal constant underlying much of modern physics.

When we say an electron moves at 1% of the speed of light, we mean:

• The speed is \( v = 0.01 \times c \) or \( v = 3 \times 10^6 \, \text{m/s} \).

This comparison is crucial in illustrating how fast particles can travel under given conditions, such as in an X-ray tube.

Protons

Protons, while also being particles like electrons, have significantly different properties.

They have a much larger mass, approximately \( 1.67 \times 10^{-27} \, \text{kg} \), compared to electrons.

• This difference in mass impacts how they respond to the same potential difference.
• Under identical potential differences as electrons, protons achieve the same kinetic energy but not the same speed.

In our example, the speed of a proton given the same potential difference is lower, around \( 2.2 \times 10^5 \, \text{m/s} \). This illustrates how particle mass influences their resultant speed when provided with the same energy.

Electrons

Electrons are fundamental particles that carry a negative electric charge and are a primary component in X-rays' production within an X-ray tube.

With a mass of \( 9.11 \times 10^{-31} \, \text{kg} \), electrons are much lighter than protons.

• Their relatively low mass allows them to be easily accelerated to high speeds under electrical fields.
• For instance, a potential difference of 255 V can accelerate electrons to reach 1% of the speed of light.

This characteristic is utilized in the X-ray tube, where electrons are accelerated from the cathode to the anode, generating X-rays as they hit the target.