A singly charged ion of unknown mass moves in a circle of radius 13.5 cm in a magnetic field of 2.9 T. The ion was accelerated through a potential difference of 3.0 kV before it entered the magnetic field. What is the mass of the ion? (hint: EPE=Vq=KE)
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what is little v and Q
To find the mass of the ion, we need to apply the principles of energy conservation and the equations related to the motion of a charged particle in a magnetic field.
Step 1: Determine the electric potential energy (EPE):
Given:
- Potential difference (V) = 3.0 kV = 3000 V
- Charge of the ion (q) = 1 (since it is singly charged)
The electric potential energy (EPE) is given by the equation EPE = Vq.
EPE = 3000 V * 1 = 3000 J
Step 2: Determine the kinetic energy (KE):
The kinetic energy (KE) of the ion is equal to the electric potential energy (EPE) since the ion was accelerated through the potential difference.
KE = EPE = 3000 J
Step 3: Determine the magnetic force (F):
The magnetic force (F) acting on a charged particle moving in a magnetic field is given by the equation F = qvB, where B is the magnetic field and v is the velocity of the particle.
In this case, the ion moves in a circle, which means there is a centripetal force acting on it. The centripetal force is provided by the magnetic force.
The centripetal force is given by the equation F = mv²/r, where m is the mass of the ion, v is its velocity, and r is the radius of the circle.
Comparing the equations F = qvB and F = mv²/r, we can equate them:
qvB = mv²/r
Step 4: Solve for the velocity (v):
Rearranging the equation, we can solve for the velocity (v):
v = qBr/m
Step 5: Calculate the mass (m):
We can substitute the values given to find the mass (m):
v = (1)(2.9 T)(0.135 m) / m
Simplifying the equation, we get:
v = 0.39215 / m
Since v appears on both sides of the equation, we can solve this quadratic equation by substituting v back into the equation:
(0.39215 / m)² = 0.39215
0.153518 / m² = 0.39215
m² = 0.153518 / 0.39215
m² = 0.3919
m = √0.3919
m ≈ 0.625 kg
Therefore, the mass of the ion is approximately 0.625 kg.
To find the mass of the ion, we can use the relationship between kinetic energy, potential energy, and charge.
First, let's determine the kinetic energy of the ion. We can use the formula for kinetic energy:
KE = 1/2 * m * v^2
Here, m represents the mass of the ion, and v represents its velocity. Since the ion is moving in a circle, we can use the relationship between velocity, radius, and time in circular motion:
v = 2πr / T
Here, r is the radius of the circular path and T is the period of the motion. Since the period is not given, we can solve for it using the equation:
T = 2πr / v
Now, let's find the period T. The period of circular motion is the time it takes for the ion to complete one full revolution. Since the ion's velocity is related to its potential energy, we can use the formula for potential energy:
PE = V * q
Here, PE represents the potential energy, V is the potential difference (3.0 kV or 3000 V), and q is the charge of the ion. The charge of the ion is given as singly charged, which means it has a charge of +1. Therefore, q = 1.
Now, we can substitute the equation for potential energy into the equation for kinetic energy:
KE = PE
1/2 * m * v^2 = V * q
Since we found the relationship between velocity and period, we can substitute that into the equation:
1/2 * m * ((2πr) / T)^2 = V * q
Simplifying, we get:
m = (V * q * T^2) / (4π^2 * r^2)
Now we can substitute the known values into the equation:
m = (3000 V * 1 * (2πr / v)^2) / (4π^2 * r^2)
The radius of the circular path given is 13.5 cm, which is equivalent to 0.135 m. The magnetic field strength is given as 2.9 T.
Now, we can find the velocity by using the formula for magnetic force:
F = q * v * B
Here, F represents the magnetic force, q is the charge of the ion, v is its velocity, and B is the magnetic field strength.
The magnetic force also has another relationship with centripetal force in circular motion:
F = m * (v^2 / r)
By equating these two forces, we can solve for the velocity:
q * v * B = m * (v^2 / r)
v = (q * B * r) / m
Now, substitute the known values into this equation:
v = (1 * 2.9 T * 0.135 m) / m
Simplifying, we get:
v = 0.3915 / m [m/s]
Now, substitute this value of velocity into the equation we derived earlier for mass:
m = (V * q * T^2) / (4π^2 * r^2)
m = (3000 V * 1 * (2πr / v)^2) / (4π^2 * r^2)
m = (3000 V * 1 * (2πr / (0.3915 / m))^2) / (4π^2 * r^2)
Simplifying, we get:
m = (3000 V * m^2 * (2πr / 0.3915)^2) / (4π^2 * r^2)
Multiplying both sides by (4π^2 * r^2), we get:
m * (4π^2 * r^2) = 3000 V * m^2 * (2πr / 0.3915)^2
Dividing both sides by m, we get:
4π^2 * r^2 = 3000 V * m * (2πr / 0.3915)^2
Dividing both sides by 2πr and simplifying, we get:
2πr = (1500 V * m) / 0.3915
Simplifying further, we get:
m = (2πr * 0.3915) / (1500 V)
Now, substitute the known values to find the mass of the ion:
m = (2π * 0.135 * 0.3915) / (1500 * 3000)
Evaluating this expression gives:
m ≈ 1.29695 x 10^-8 kg
Therefore, the mass of the ion is approximately 1.29695 x 10^-8 kg.
Vq=1/2 m v^2
v= sqrt 2 Vq/m
centripetal force=magneticforce
1/2 m v^2/r= Bqv
solve for mass.
solve for m.