Solved Quantum Physics Worksheet 1) What is the direction of ... - Free Printable

Problem 1: Direction of the External Magnetic Field

#### Question:
What is the direction of the external magnetic field in the diagram below? Explain.

#### Diagram Description:
- Gamma rays are incident on a lead plate.
- Electrons (\(e^-\)) and positrons (\(e^+\)) are emitted from the lead plate.
- The trajectories of the electrons and positrons are curved, indicating the presence of a magnetic field.

#### Solution:
The curvature of the trajectories of charged particles in a magnetic field is governed by the Lorentz force law:
\[
\mathbf{F} = q(\mathbf{v} \times \mathbf{B})
\]
where:
- \(q\) is the charge of the particle,
- \(\mathbf{v}\) is the velocity of the particle,
- \(\mathbf{B}\) is the magnetic field vector.

The direction of the force is determined by the right-hand rule:
- If you point your right hand's thumb in the direction of the velocity (\(\mathbf{v}\)),
- and your fingers in the direction of the magnetic field (\(\mathbf{B}\)),
- then your palm will point in the direction of the force (\(\mathbf{F}\)).

From the diagram:
- Electrons (\(e^-\)) are negatively charged (\(q < 0\)).
- Positrons (\(e^+\)) are positively charged (\(q > 0\)).
- Both particles are deflected in opposite directions, which indicates that the magnetic field is perpendicular to their velocities.

By analyzing the deflection:
- Electrons (\(e^-\)) are deflected downward.
- Positrons (\(e^+\)) are deflected upward.

Using the right-hand rule:
- For electrons (\(e^-\)):
- The velocity is to the right.
- The force is downward.
- Therefore, the magnetic field must be into the page (perpendicular to both the velocity and the force).
- For positrons (\(e^+\)):
- The velocity is to the right.
- The force is upward.
- Again, the magnetic field must be into the page.

Thus, the direction of the external magnetic field is into the page.

#### Final Answer:
\[
\boxed{\text{Into the page}}
\]

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Problem 2: Infinite Range of Electromagnetic and Gravitational Interactions

#### Question:
What does the infinite range of the electromagnetic and gravitational interactions tell you about the masses of the photon and the graviton? Explain.

#### Solution:
The range of a fundamental force is related to the mass of the particle that mediates the force. This relationship is described by the Yukawa potential, which states that the strength of the force decreases exponentially with distance:
\[
V(r) \propto \frac{e^{-mr}}{r}
\]
where:
- \(m\) is the mass of the mediating particle,
- \(r\) is the distance between the interacting particles.

If the mediating particle has a non-zero mass (\(m > 0\)), the exponential term \(e^{-mr}\) causes the force to decrease rapidly with distance, resulting in a finite range. Conversely, if the mediating particle is massless (\(m = 0\)), the exponential term becomes 1, and the force follows an inverse-square law (\(\frac{1}{r}\)), leading to an infinite range.

#### Electromagnetic Force:
- The mediator of the electromagnetic force is the photon.
- The electromagnetic force has an infinite range, as observed in phenomena like light propagation over vast distances.
- Therefore, the photon must be massless (\(m_\gamma = 0\)).

#### Gravitational Force:
- The mediator of the gravitational force is hypothesized to be the graviton.
- The gravitational force also has an infinite range, as observed in phenomena like planetary orbits and galaxy dynamics.
- Therefore, the graviton is also hypothesized to be massless (\(m_g = 0\)).

#### Final Answer:
\[
\boxed{\text{The infinite range of electromagnetic and gravitational interactions implies that the photon and the graviton are massless.}}
\]

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Problem 3: Stopping at the Footpath Instead of Falling Through the Earth

#### Question:
When you jump off a building, why do you stop falling at the footpath rather than continuing through to the center of the Earth? Explain in terms of gravitational and electromagnetic forces.

#### Solution:
When you jump off a building, two primary forces come into play: gravitational force and electromagnetic force.

1. Gravitational Force:
- As you fall, the Earth's gravitational force pulls you downward. This force accelerates you toward the ground.
- When you reach the footpath, the gravitational force continues to act, but it is no longer the dominant force because of the interaction with the footpath.

2. Electromagnetic Force:
- The footpath is composed of atoms, which have nuclei and electrons. These atoms are held together by strong nuclear forces within the nuclei and electromagnetic forces between the electrons.
- When you collide with the footpath, your body's atoms interact with the atoms of the footpath. The repulsive electromagnetic forces between the electrons in your body and the electrons in the footpath prevent you from passing through the material.
- These repulsive forces are extremely strong at short distances and effectively stop your motion.

3. Why You Don't Fall Through the Earth:
- If you were to somehow pass through the footpath (ignoring the electromagnetic forces), you would continue to fall due to gravity until you reached the center of the Earth.
- However, the electromagnetic forces between atoms ensure that this does not happen. The footpath provides a solid barrier that stops your fall.

#### Final Answer:
\[
\boxed{\text{You stop falling at the footpath because the repulsive electromagnetic forces between the atoms in your body and the atoms in the footpath prevent you from passing through the material.}}
\]