A beam of ∝ particles (helium nuclei) is used to treat a tumor located 10.0 cm inside a patient. To penetrate to the tumor, the ∝ particles must be accelerated to a speed of 0.458c, where c is the speed of light. (Ignore relativistic effects?
Answers
Lorentz Force equation of charged particle
q
moving with a velocity
v
in an Electric field
→
E
and magnetic field
→
B
is given as
→
F
=
q
(
→
E
+
→
v
×
→
B
)
Considering the magnetic field which is to be found out.
In a cyclotron, magnetic field
→
B
is perpendicular to the velocity
→
v
of charge
q
.
Hence,
∣
∣
∣
→
F
∣
∣
∣
=
q
∣
∣
→
v
∣
∣
∣
∣
∣
→
B
∣
∣
∣
⇒
∣
∣
∣
→
B
∣
∣
∣
=
∣
∣
∣
→
F
∣
∣
∣
q
∣
∣
→
v
∣
∣
........(1)
As the
α
particles move in a circular field of radius
r
these experience centripetal force provided by the magnetic field.
∣
∣
∣
→
F
∣
∣
∣
=
m
v
2
r
.........(2)
substituting value of force from (2) in 91) we get
∣
∣
∣
→
B
∣
∣
∣
=
m
v
2
r
q
∣
∣
→
v
∣
∣
=>|vecB|= (m|vecv|)/(qr)
Using m_alpha = 6.644xx10^-27 kg, |vecv| = 0.458c, r = 1.00 m, and q = 3.204 xx 10^-19 C for "He"^(2+) we get
|vecB|= (6.644xx10^-27xx0.458xx 2.998xx10^8)/(3.204 xx 10^-19xx1.00)
|vecB|= 2.87T