To compute the force acting on the proton, we use the formula for the Lorentz force:

$\mathbf{F} = q \mathbf{v} \times \mathbf{B}$

Where:

• $q = 1.6 \times 10^{-19} \, \text{C}$ is the charge of the proton.
• $\mathbf{v} = 2 \times 10^5 \hat{k} \, \text{m/s}$ is the velocity of the proton.
• $\mathbf{B} = 0.5 \hat{i} \, \text{T}$ is the magnetic field.

We need to compute the cross product $\mathbf{v} \times \mathbf{B}$. Since the velocity vector is along the $\hat{k}$-direction and the magnetic field is along the $\hat{i}$-direction, the cross product follows the right-hand rule.

$\mathbf{v} \times \mathbf{B} = (2 \times 10^5 \hat{k}) \times (0.5 \hat{i})$

Using the properties of the cross product: $\hat{k} \times \hat{i} = -\hat{j}$

So, $\mathbf{v} \times \mathbf{B} = 2 \times 10^5 \times 0.5 \times (-\hat{j}) = -10^5 \hat{j} \, \text{m/s}$

Now, calculate the force: $\mathbf{F} = q \times \mathbf{v} \times \mathbf{B} = (1.6 \times 10^{-19} \, \text{C}) \times (-10^5 \hat{j} \, \text{m/s})$

$\mathbf{F} = -1.6 \times 10^{-14} \hat{j} \, \text{N}$

Thus, the force acting on the proton is:

$\mathbf{F} = -1.6 \times 10^{-14} \hat{j} \, \text{N}$

This means the force is directed along the negative $\hat{j}$-axis.

Would you like more details or further clarifications on any step?

Here are some related questions:

1. What is the right-hand rule for determining the direction of a cross product in physics?
2. How does the velocity of a charged particle in a magnetic field affect the magnitude of the force?
3. What is the significance of the charge sign (positive or negative) in determining the direction of the Lorentz force?
4. How would the force change if the magnetic field had a different direction or magnitude?
5. How does the speed of the particle affect the Lorentz force in practical scenarios like in particle accelerators?

Tip: To visualize the direction of the force using the right-hand rule, curl your fingers in the direction of $\mathbf{v}$ and curl them toward $\mathbf{B}$; your thumb will point in the direction of the force.