To determine the magnetic force on a proton moving through a magnetic field, we will use the formula for the magnetic force on a charged particle:
[tex]\[ F = q \cdot v \cdot B \cdot \sin(\theta) \][/tex]
where:
- [tex]\( F \)[/tex] is the magnetic force,
- [tex]\( q \)[/tex] is the charge of the proton,
- [tex]\( v \)[/tex] is the velocity of the proton,
- [tex]\( B \)[/tex] is the magnetic field strength, and
- [tex]\( \theta \)[/tex] is the angle between the velocity and the magnetic field direction.
### Step-by-Step Solution:
1. Given Values:
- Velocity, [tex]\( v = 2.5 \times 10^7 \)[/tex] m/s
- Magnetic field, [tex]\( B = 3.4 \)[/tex] T
- Charge of a proton, [tex]\( q = 1.6 \times 10^{-19} \)[/tex] C
- The angle [tex]\( \theta \)[/tex] between the velocity and the magnetic field [tex]\( = 90^\circ \)[/tex] (since the velocity of the proton is to the left, and the magnetic field is pointing toward you)
2. Calculate [tex]\( \sin(\theta) \)[/tex]:
- Since [tex]\( \theta = 90^\circ \)[/tex], [tex]\( \sin(90^\circ) = 1 \)[/tex].
3. Substitute the values into the formula:
[tex]\[
F = 1.6 \times 10^{-19} \, \text{C} \times 2.5 \times 10^7 \, \text{m/s} \times 3.4 \, \text{T} \times 1
\][/tex]
4. Perform the multiplication:
[tex]\[
F = 1.6 \times 2.5 \times 10^{-19 + 7} \times 3.4
\][/tex]
[tex]\[
F = 1.6 \times 2.5 \times 3.4 \times 10^{-12}
\][/tex]
[tex]\[
F \approx 13.6 \times 10^{-12} \, \text{N}
\][/tex]
[tex]\[
F \approx 1.36 \times 10^{-11} \, \text{N}
\][/tex]
5. Determine the direction of the force:
- Use the right-hand rule: Point your right hand’s fingers in the direction of the velocity (to the left). Point your thumb in the direction of the magnetic field (toward you). Your palm faces the direction of the force on a positive charge.
- With this setup, the force is directed upward.
6. Match the solution with the multiple-choice options:
- The calculated force magnitude is [tex]\( 1.36 \times 10^{-11} \, \text{N} \)[/tex].
- The direction is upward.
Thus, the correct answer is:
[tex]\[ \boxed{\text{C. } 1.4 \times 10^{-11} \text{ N up}} \][/tex]
