Answer :
Answer:
diameter = 0.640 m
Explanation:
We can find the diameter of the orbit by using this principle:
[tex]\boxed{F_{centripetal}=F_{magnet}}[/tex]
(since the centripetal force is caused by the magnetic force)
The formulas for centripetal force & magnetic force are:
[tex]\boxed{F_{centripetal}=\frac{mv^2}{r} }[/tex]
[tex]\boxed{F_{magnetic}=qvB}[/tex]
where:
- [tex]m=\text{mass}[/tex]
- [tex]v=\text{velocity}[/tex]
- [tex]r=\text{radius}[/tex]
- [tex]q=\text{charge}[/tex]
- [tex]B=\text{magnetic field}[/tex]
Given:
- [tex]m=2.84\times10^{-26}\ kg[/tex]
- [tex]v=3.65\%\times3.00\times10^8=1.095\times10^7\ m/s[/tex]
- [tex]q=1.602\times10^{-19}\ C[/tex]
- [tex]B=0.936\ T[/tex]
Then:
[tex]\begin{aligned}F_{centripetal}&=F_{magnet}\\\frac{mv^2}{r}&=qvB \\\frac{mv}{r} &=qB\\r&=\frac{qB}{mv} \\\\&=\frac{1.062\times10^{-19}\times0.936}{2.84\times10^{-26}\times1.095\times10^7} \\\\&=\frac{1.062\times0.936}{2.84\times1.095} \times10^{(-19+26-7)}\\\\&=\bf0.320\ m\end{aligned}[/tex]
By using the radius, we can find the diameter:
[tex]\begin{aligned}diameter&=2\times radius\\&=2\times0.320\times10^{-1}\\&=\bf0.640\ m\end{aligned}[/tex]
The diameter of the orbit is 4.15 meters.
To find the diameter of the orbit in a cyclotron, we need to use the relationship between the centripetal force and the magnetic force on the ions. The centripetal force required to keep an ion in circular motion is provided by the magnetic force.
The formula for the magnetic force on a charged particle moving in a magnetic field is given by:
F = q × v × B
where:
F is the magnetic force,
q is the charge of the ion,
v is the speed of the ion,
B is the magnetic field.
The centripetal force required for circular motion is given by:
- [tex]\[F = \frac{mv^2}{r}\][/tex]
Where:
- m is the mass of the ion,
- v is the speed of the ion,
- r is the radius of the orbit.
Setting these two expressions for the force equal to each other:
- q × v × B = [tex]\frac{mv^2}{r}[/tex]
Solving for r:
- [tex]\[r = \frac{mv}{qB}\][/tex]
Given:
B = 0.936 T
m = [tex]2.84 \times 10^{-26} \, \text{kg} \\[/tex]
v = [tex]0.0365 \times 3.00 \times 10^8 \, \text{m/s} = 1.095 \times 10^7 \, \text{m/s} \\[/tex]
q = [tex]1.6 \times 10^{-19} \, \text{C}[/tex]
Plugging in these values:
- [tex]\[r = \frac{(2.84 \times 10^{-26} \, \text{kg})(1.095 \times 10^7 \, \text{m/s})}{(1.6 \times 10^{-19} \, \text{C})(0.936 \, \text{T})}\][/tex]
- [tex]\[r = \frac{3.1074 \times 10^{-19}}{1.4976 \times 10^{-19}}\][/tex]
- [tex]\[r \approx 2.075 \, \text{m}\][/tex]
The diameter (d) is twice the radius:
- [tex]\[d = 2r[/tex]
- [tex]= 2 \times 2.075 \, \text{m}[/tex]
- [tex]= 4.15 \, \text{m}[/tex]
So, the calculated diameter is 4.15 meters.
