To determine the amount of the charge, we will use the formula for the magnetic force experienced by a moving charge in a magnetic field:
[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.
- [tex]\( v \)[/tex] is the velocity of the charge.
- [tex]\( B \)[/tex] is the magnetic field.
- [tex]\( \theta \)[/tex] is the angle between the velocity and the magnetic field.
We need to solve for [tex]\( q \)[/tex]:
[tex]\[ q = \frac{F}{v \cdot B \cdot \sin(\theta)} \][/tex]
Given values:
- Force, [tex]\( F = 7.89 \times 10^{-7} \)[/tex] N
- Velocity, [tex]\( v = 2090 \)[/tex] m/s
- Angle, [tex]\( \theta = 29.4^\circ \)[/tex]
- Magnetic field, [tex]\( B = 4.23 \times 10^{-3} \)[/tex] T
Step-by-step solution:
1. Convert the angle from degrees to radians:
[tex]\[ \theta = 29.4^\circ \][/tex]
[tex]\[ \theta_{\text{rad}} = \frac{29.4 \times \pi}{180} \][/tex]
2. Calculate the sine of the angle in radians:
[tex]\[ \sin(\theta_{\text{rad}}) = \sin\left(\frac{29.4 \times \pi}{180}\right) \][/tex]
3. Substitute all values into the formula to solve for [tex]\( q \)[/tex]:
[tex]\[ q = \frac{7.89 \times 10^{-7} \, \text{N}}{2090 \, \text{m/s} \cdot 4.23 \times 10^{-3} \, \text{T} \cdot \sin\left(\frac{29.4 \times \pi}{180}\right)} \][/tex]
Now, simplifying this will give a numerical value for the charge [tex]\( q \)[/tex].
Upon calculation, we find that the charge [tex]\( q \)[/tex] can be expressed in scientific notation as:
[tex]\[ q = 1.8180004938057521 \times 10^{-7} \, \text{C} \][/tex]
Thus, the amount of the charge is:
[tex]\[ 1.818 \times 10^{-7} \, \text{C} \][/tex]
So, the charge is approximately:
[tex]\[ 1.818 \times 10^{-7} \, \text{C} \][/tex]
