To determine the force exerted on a moving charge in a magnetic field, we use the formula for the magnetic force on a charged particle:
[tex]\[ F = |q| \cdot v \cdot B \cdot \sin(\theta) \][/tex]
Here, the symbols represent the following:
- [tex]\( F \)[/tex] is the force exerted on the charge.
- [tex]\( q \)[/tex] is the charge.
- [tex]\( v \)[/tex] is the velocity of the charge.
- [tex]\( B \)[/tex] is the magnetic field strength.
- [tex]\( \theta \)[/tex] is the angle between the velocity direction and the magnetic field.
Given values:
- Charge, [tex]\( q = -2.0 \mu C = -2.0 \times 10^{-6} \, \text{C} \)[/tex]
- Velocity, [tex]\( v = 5.0 \times 10^6 \, \text{m/s} \)[/tex]
- Magnetic field, [tex]\( B = 3.0 \times 10^{-4} \, \text{T} \)[/tex]
- Angle, [tex]\( \theta = 20^{\circ} \)[/tex]
Step-by-step solution:
1. Convert the angle from degrees to radians:
[tex]\[
\theta_{\text{radians}} = \theta_{\text{degrees}} \cdot \frac{\pi}{180}
\][/tex]
[tex]\[
\theta_{\text{radians}} = 20 \cdot \frac{\pi}{180} \approx 0.3491 \, \text{radians}
\][/tex]
2. Substitute the given values into the formula for the force:
[tex]\[
F = |q| \cdot v \cdot B \cdot \sin(\theta_{\text{radians}})
\][/tex]
- Absolute value of charge, [tex]\( |q| = 2.0 \times 10^{-6} \, \text{C} \)[/tex]
- Velocity, [tex]\( v = 5.0 \times 10^6 \, \text{m/s} \)[/tex]
- Magnetic field, [tex]\( B = 3.0 \times 10^{-4} \, \text{T} \)[/tex]
- Sine of the angle, [tex]\( \sin(0.3491) \approx 0.342 \)[/tex]
3. Calculate the force:
[tex]\[
F = (2.0 \times 10^{-6} \, \text{C}) \cdot (5.0 \times 10^6 \, \text{m/s}) \cdot (3.0 \times 10^{-4} \, \text{T}) \cdot 0.342
\][/tex]
4. Perform the multiplication:
[tex]\[
F \approx 2.0 \times 5.0 \times 3.0 \times 0.342 \times 10^{-6} \times 10^6 \times 10^{-4}
\][/tex]
[tex]\[
F \approx 30 \times 0.342 \times 10^{-4}
\][/tex]
[tex]\[
F \approx 10.26 \times 10^{-4} \, \text{N}
\][/tex]
[tex]\[
F \approx 0.001026 \, \text{N}
\][/tex]
5. Determine the closest matching result in scientific notation:
The force exerted on the moving charge is [tex]\( 0.001026 \, \text{N} \)[/tex]. This value matches [tex]\( 1.0 \times 10^{-3} \, \text{N} \)[/tex] from the given options.
Answer:
The force exerted on the moving charge is [tex]\( \mathbf{1.0 \times 10^{-3} \, N} \)[/tex].
