An alpha particle is accelerated by a voltage of 1.53*10^3 V and is then deflected by a magnetic field of strength 0.020 T into a circular path of radius 0.40 m. If the alpha particles have a charge of 3.2*10^-19 C, what is their mass?
To find the mass of the alpha particle, we can use the principles of circular motion and the relationship between the centripetal force and the magnetic force experienced by a charged particle moving through a magnetic field.
Given:
Voltage (V) = 1.53 * 10^3 V
Magnetic field strength (B) = 0.020 T
Radius (r) = 0.40 m
Charge (q) = 3.2 * 10^-19 C
1. Calculate the velocity of the alpha particle using the voltage:
The voltage can be related to the kinetic energy of the particle using the equation: qV = 1/2mv^2, where q is the charge, V is the voltage, m is the mass, and v is the velocity.
Rearranging the equation to solve for velocity:
v = √((2qV) / m)
2. Calculate the acceleration of the alpha particle:
The acceleration of a charged particle in a magnetic field is given by the equation: a = qvB / m, where a is the acceleration, q is the charge, v is the velocity, B is the magnetic field strength, and m is the mass.
Rearranging the equation to solve for mass:
m = qvB / a
3. Calculate the centripetal acceleration of the alpha particle:
The centripetal acceleration of a particle moving in a circular path is given by the equation: a = v^2 / r, where a is the acceleration, v is the velocity, and r is the radius.
Substitute the value of centripetal acceleration into the equation above:
m = qvB / (v^2 / r)
4. Simplify the equation:
m = qBr / v
5. Substitute the given values into the equation:
m = (3.2 * 10^-19 C)(0.020 T)(0.40 m) / v
6. Substitute the value of v from step 1 into the equation:
m = (3.2 * 10^-19 C)(0.020 T)(0.40 m) / √((2 × 3.2 × 10^-19 C × 1.53 × 10^3 V) / m)
7. Rearrange and simplify the equation to solve for mass:
m^2 = (3.2 * 10^-19 C)(0.020 T)(0.40 m) / (2 × 3.2 × 10^-19 C × 1.53 × 10^3 V)
m^2 = (0.256 * 10^-19 T·C·m^2) / (9.78 × 10^-19 V)
m^2 = 0.0262 kg
m ≈ √0.0262 kg
m ≈ 0.162 kg
Therefore, the mass of the alpha particle is approximately 0.162 kg.
To find the mass of the alpha particles, we can use the principles of circular motion in a magnetic field. The centripetal force acting on the particles is provided by the magnetic force, which is given by the equation:
F_mag = q * v * B
Where:
F_mag 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 is given by:
F_cen = (m * v^2) / r
Where:
F_cen is the centripetal force
m is the mass of the particle
v is the velocity of the particle
r is the radius of the circular path
Since the centripetal force and the magnetic force are equal, we can equate the two equations:
(q * v * B) = (m * v^2) / r
We can simplify this equation by canceling out the velocity (v) term:
(q * B) = (m * v) / r
Rearranging the equation to solve for mass (m):
m = (q * B * r) / v
Now we can substitute the given values into the equation:
q = 3.2 * 10^-19 C
B = 0.020 T
r = 0.40 m
To find the velocity (v), we need to consider the relationship between voltage (V) and kinetic energy (KE) of the particles. The kinetic energy can be expressed as:
KE = (1/2) * m * v^2
The voltage is used to accelerate the particles, which can be equated to the change in kinetic energy:
V = KE / q
Rearranging this equation to solve for velocity (v):
v = sqrt((2 * q * V) / m)
Since we know the voltage (V) is 1.53 * 10^3 V, we can substitute the values and solve for velocity (v).
Now we substitute the values of charge (q), magnetic field strength (B), radius (r), and velocity (v) into the mass equation to find the mass (m).