A positively charged particle of mass 5.60 10-8 kg is traveling due east with a speed of 60 m/s and enters a 0.49-T uniform magnetic field. The particle moves through one-quarter of a circle in a time of 1.20 10-3 s, at which time it leaves the field heading due south. All during the motion the particle moves perpendicular to the magnetic field.
To find the charge of the particle, you need to use the formula for the centripetal force experienced by a charged particle moving in a magnetic field. The centripetal force is provided by the magnetic force acting on the charged particle.
The formula for the magnetic force on a charged particle moving through a magnetic field is given by:
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
In this case, we are given the values for the mass of the particle (m = 5.60 x 10^-8 kg), the speed of the particle (v = 60 m/s), and the magnetic field strength (B = 0.49 T). We need to find the charge of the particle (q).
First, we need to find the magnitude of the magnetic force experienced by the particle. The magnetic force is given by the formula:
F = m * a
Where:
- F is the force
- m is the mass
- a is the acceleration
We know that the acceleration is centripetal and is given by the formula:
a = (v^2) / r
Where:
- a is the acceleration
- v is the velocity
- r is the radius of the circular path
In this case, we are given the time taken to complete one-quarter of a circle (t = 1.20 x 10^-3 s). Since the particle moves through one-quarter of a circle, the angle covered is 90 degrees or π/2 radians. Therefore, the time taken to complete one full circle (T) is given by:
T = 4 * t
To find the radius of the circular path (r), we can use the formula:
r = (m * v) / (q * B)
Now we can substitute the values into the equations:
T = 4 * t = 4 * 1.20 x 10^-3 s = 4.80 x 10^-3 s
r = (m * v) / (q * B)
Next, rearrange the equation to solve for q:
q = (m * v) / (r * B)
Substitute the known values into the equation:
q = (5.60 x 10^-8 kg * 60 m/s) / [(r * 0.49 T)]
The value of r can be found using the equation for the circumference of a circle:
C = 2πr
Since the particle moves through one-quarter of a circle, the distance covered is given by one-quarter of the circumference:
C = πr / 4
To find r, rearrange the equation:
r = 4C / π
Substitute the known values into the equation:
r = (4 * [2πr]) / π = 8r
Now we can substitute the value of r into the equation for q:
q = (5.60 x 10^-8 kg * 60 m/s) / [(8 * r * 0.49 T)]
Now substitute T = 4.80 x 10^-3 s and solve for q.