A particle with a mass of 2.00x10-16 kg and a charge of 22.0 nC starts from rest, is accelerated by a strong electric field, and is fired from a small source inside a region of uniform constant magnetic field 0.660 T. The velocity of the particle is perpendicular to the field. The circular orbit of the particle encloses a magnetic flux of 15.0 µWb.
(a) Calculate the speed of the particle.
m/s
(b) Calculate the potential difference (ΔV) through which the particle accelerated inside the source.
V
To calculate the speed of the particle, we can use the concept of the magnetic force and the centripetal force acting on a charged particle moving in a magnetic field.
The magnetic force (F) on a charged particle moving in a magnetic field is given by the equation:
F = q * v * B
Where:
- F is the magnetic force
- q is the charge of the particle
- v is the velocity of the particle
- B is the magnetic field strength
The centripetal force (Fc) acting on a charged particle moving in a circle is given by the equation:
Fc = (m * v^2) / R
Where:
- m is the mass of the particle
- v is the velocity of the particle
- R is the radius of the circular path
In this case, we know that the circular orbit of the particle encloses a magnetic flux of 15.0 µWb.
The magnetic flux (Φ) is given by the equation:
Φ = B * A
Where:
- Φ is the magnetic flux
- B is the magnetic field strength
- A is the area enclosed by the circular path
Since the magnetic field strength and the area are given, we can calculate the radius (R) of the circular path.
R = sqrt(Φ / (pi * B))
Now, equating the magnetic force with the centripetal force:
q * v * B = (m * v^2) / R
Simplifying this equation, we can solve for the velocity (v):
v = (q * B * R) / m
Let's substitute the given values and calculate the speed of the particle:
q = 22.0 nC = 22.0 * 10^-9 C
m = 2.00 * 10^-16 kg
B = 0.660 T
Φ = 15.0 µWb = 15.0 * 10^-6 Wb
R = sqrt((15.0 * 10^-6) / (pi * 0.660))
Now substitute the value of R into the velocity equation:
v = (22.0 * 10^-9 * 0.660 * R) / (2.00 * 10^-16)
Evaluate the expressions and calculate the speed of the particle.
To calculate the potential difference (ΔV), we can use the equation for the work done on a charged particle in an electric field:
ΔV = W / q
Where:
- ΔV is the potential difference
- W is the work done
- q is the charge of the particle
In this case, the particle starts from rest, so the initial velocity (vi) is zero. The final velocity (vf) is the calculated speed of the particle.
The work done (W) on the particle is equal to the change in kinetic energy:
W = (1/2) * m * (vf^2 - vi^2)
Since vi = 0, the equation simplifies to:
W = (1/2) * m * vf^2
Now, substituting the values and calculating the potential difference:
m = 2.00 * 10^-16 kg
vf = (calculated speed from part a)
ΔV = (1/2) * (2.00 * 10^-16) * (vf^2)
Evaluate the expression and calculate the potential difference (ΔV).