The mass of an atom can be determined by ionizing it and accelerating it through a known electric field followed by a known magnetic field. The ion’s circular path is then measured. Calculate the mass of a chlorine anion of charge 3.20 × 10−19 C if it follows a circular path of radius 0.862 cm after it has been accelerated through a potential difference of 2.5 × 102 V and the exposed to a uniform magnetic field of strength 3.00 T.
To calculate the mass of the chlorine anion, we can use the formula for the centripetal force acting on a charged particle moving in a magnetic field.
The centripetal force is given by the equation:
F = B * q * v
Where F is the centripetal force, B is the magnetic field strength, q is the charge of the particle, and v is the velocity of the particle.
In this case, the force is provided by the electric field formed due to the accelerated ion, and it is given by the equation:
F = q * E
Where E is the electric field strength and q is the charge of the ion.
Since the force is the same in both cases, we can set these two equations equal to each other:
B * q * v = q * E
The charge of the ion is given as 3.20 × 10^−19 C, and the potential difference is given as 2.5 × 10^2 V.
Using the formula for electric potential difference:
V = E * d
Where V is the potential difference, E is the electric field strength, and d is the distance the ion traveled.
Rearranging the equation above, we can find the electric field strength:
E = V / d
Given that the distance is 0.862 cm, which is 0.00862 m, and the potential difference is 2.5 × 10^2 V, we can substitute these values into the equation to find the electric field strength.
E = 2.5 × 10^2 V / 0.00862 m
Next, we need to find the velocity of the ion, which can be determined from the radius of the circular path.
The centripetal force is given by the equation:
F = m * v^2 / r
Where F is the centripetal force, m is the mass of the ion, v is the velocity of the ion, and r is the radius of the circular path.
However, in this case, we already have the force given as B * q * v.
Setting these two equations equal to each other:
B * q * v = m * v^2 / r
Simplifying the equation:
B * q = m * v / r
Therefore, the velocity of the ion can be calculated as:
v = B * q * r / m
Now, we have enough information to solve for the mass.
1. Calculate the electric field strength using the given potential difference and distance:
E = 2.5 × 10^2 V / 0.00862 m
2. Calculate the velocity of the ion using the magnetic field strength, charge, radius, and mass (which is the unknown we are solving for):
v = (3.00 T) * (3.20 × 10^−19 C) * (0.00862 m) / m
3. Set the equations for the centripetal force and electric force equal to each other:
B * q * v = q * E
4. Solve for the mass:
B * q * v = q * E
(3.00 T) * (3.20 × 10^−19 C) * (0.00862 m) = (3.20 × 10^−19 C) * E
m = (3.00 T) * (3.20 × 10^−19 C) * (0.00862 m) / E
By substituting the calculated values for E, B, q, v, and radius, you can find the value of the mass of the chlorine anion.