A singly charged positive ion has a mass of 3.20 10-26 kg. After being accelerated from rest through a potential difference of 823 V, the ion enters a magnetic field of 0.820 T along a direction perpendicular to the direction of the field. Calculate the radius of the path of the ion in the field.
cm
To find the radius of the path of the ion, we can use the principles of circular motion and combine the equations for centripetal force and magnetic force.
First, let's find the velocity of the ion after being accelerated through the potential difference.
The potential difference (V) is given as 823 V. The electric potential energy gained by the ion can be calculated using the equation:
Potential Energy (PE) = charge (q) × potential difference (V)
Since the ion is singly charged, the charge (q) is equal to +1.6 × 10^-19 C (coulombs). Substituting the values, we can find the potential energy gained by the ion.
PE = (+1.6 × 10^-19 C) × (823 V)
Next, we'll equate the potential energy gained to the kinetic energy gained by the ion after being accelerated.
Kinetic Energy (KE) = 1/2 × mass (m) × velocity squared (v^2)
The mass of the ion (m) is given as 3.20 × 10^-26 kg. Rearranging the equation, we can solve for velocity (v).
v^2 = (2 × PE) / m
Substituting the values:
v^2 = (2 × PE) / (3.20 × 10^-26 kg)
Now, we can find the velocity (v) by taking the square root of both sides of the equation.
v = √[(2 × PE) / (3.20 × 10^-26 kg)]
Moving on, we can calculate the radius (r) using the equation for the magnetic force experienced by a charged particle moving perpendicularly to a magnetic field.
Magnetic Force (F) = charge (q) × velocity (v) × magnetic field strength (B)
The magnetic field strength (B) is given as 0.820 T (teslas), and the charge (q) is +1.6 × 10^-19 C. Rearranging the equation for radius (r), we have:
r = (m × v) / (q × B)
Substituting the values, we can find the radius of the path of the ion in centimeters.
r = ((3.20 × 10^-26 kg) × (√[(2 × PE) / (3.20 × 10^-26 kg)])) / ((+1.6 × 10^-19 C) × (0.820 T))
By solving this equation, we get the value of r in meters. To convert it to centimeters, we multiply by 100, as there are 100 centimeters in a meter.