Answer:
1.64 T direction of the magnetic field is into the page .
Explanation:
Solution
The magnetic force F_B causes the a-particle to move in a circular motion where the a-particle gains a centrifugal force F_C So, the magnetic force and the centrifugal force are equal to each other.
F_B = F_C (1)
The magnetic force is affected by the magnitude and the direction of the magnetic force, the direction of the magnetic field, the sign of the charge and the direction of the moving of the charge and it is given by
F_B = qvB
The centrifugal force is related to the mass of the a-particle ma and the circular radius R by
F_C = m_a*v^2/R
Now let us plug the expressions of F_C and F_B into equation (1) to get the magnetic field B
F_B = F_C
qvB =m_a*v^2/R (solve for B)
B = m_a*v^2/qR (2)
The term {v/R) equals the angular frequency which equals 2[tex]\pi[/tex]/T . From the next steps, we got this relationship
v = x/T = 2[tex]\pi[/tex]R/T
v/R = 2[tex]\pi[/tex]/T
Where the distance x equals the circumference of the circle where the alpha particle moves. Hence, the magnetic field is given by
B = (m_a/q)*(v/R)
= (m_a/q)*2[tex]\pi[/tex]/T (3)
Where T is the time, mo, is the mass of the alpha particle and q is the charge of the alpha particle and equals 2e.
Alpha particle consists of two protons and two neutrons, therefore, the total mass of the alpha particle is given by
m_a = 2m_p+ 2m_n
= 2 (1.672 x 10^-27 kg) +2 (1.675 x 10^-27 kg)
= 6.7 x 10^-27 kg
Now we can plug our values for m_a,q and T into equation (3) to get the magnetic field B
B = m_a*2[tex]\pi[/tex]/q*T
= 2[tex]\pi[/tex](6.7*10^-27)/2(1.6*10^-19)(81*10^-9s)
= 1.64 T
The right-hand rule determines the direction of the magnetic field B and the direction of the motion of the alpha particle. Where your thump is in the direction of the magnetic field, while your remaining fingers curl in the direction of the motion of the alpha particle (Clockwise). Let us apply this rule, we find that the direction of the magnetic field is into the page .
This question involves the concepts of centripetal force, right-hand rule, and magnetic force.
The magnetic field is found to be "1.61 T (into the page)".
In order for the alpha particle to move in a circular path, the centripetal force must be equal to the magnetic force.
[tex]Magnetic\ Force = Centripetal\ Force\\\\qvB=\frac{mv^2}{r}\\\\B=\frac{mv}{qr}\\\\B=\frac{mr\omega}{qr}\\\\B=\frac{m\omega}{q}\\\\[/tex]
where,
B = magnetic field of coil = ?
m = mass of alpha particle = 6.644 x 10⁻²⁷ kg
v = tangential speed = [tex]\frac{2\pi r}{t}[/tex]
t = 81 ns = 8.1 x 10⁻⁸ s
ω = angular speed = [tex]\frac{v}{r}=\frac{2\pi r}{rt}=\frac{2\pi}{8.1\ x\ 10^{-8}\ s}[/tex] = 7.75 x 10⁷ rad/s
q = charge on alpa particle = 3.2 x 10⁻¹⁹ C
Therefore,
[tex]B=\frac{(6.644\ x\ 10^{-27}\ kg)(7.75\ x\ 10^7\ rad/s)}{3.2\ x\ 10^{-19}\ C}[/tex]
B = 1.61 T
Now, we will use the right-hand rule to find out the direction of the magnetic field. The right-hand rule states that when you curl the fingers of the right hand in the direction of motion of the particle, the direction of the thumb gives the direction of the magnetic field.
Applying the right-hand rule in this scenario gives the direction of the magnetic field into the page.
Learn more about magnetic force here:
https://brainly.com/question/12824331?referrer=searchResults
The attached picture shows the right-hand rule.
