A particle of mass 7.7 × 10-8 kg and charge +11 μC is traveling due east. It enters perpendicularly a magnetic field whose magnitude is 4.2 T. After entering the field, the particle completes one-half of a circle and exits the field traveling due west. How much time does the particle spend in the magnetic field?
The force from the magnetic field equals centripetal force
Bqv=mv^2/r
v=Bqr/m
but PI*r/T=V so
PI*r/T=Bqr/m
Time=PI*m/Bq
check that.
Well, isn't this particle quite the show-off! Going in one end and coming out the other, like a circus clown shooting out of a cannon! But let's crunch some numbers and figure out how long this spectacle takes.
We know that the force on a charged particle moving in a magnetic field is given by the equation F = qvB, where F is the force, q is the charge, v is the velocity, and B is the magnetic field strength.
The force on our particle is centripetal, because it's moving in a circle. So, we can equate the magnetic force to the centripetal force:
qvB = mv²/r
Where m is the mass of the particle, v is its velocity, and r is the radius of the circular path.
Now, we know from the question that the particle does half a circle in the magnetic field. So the radius of its path is half the circumference. We also know that the magnitude of the centripetal force is provided by the weight of the particle itself. In other words, gravity balances out the magnetic force.
Thus, we can write:
qvB = mg
Rearranging the equation for the velocity, we get:
v = (mgr)/(qB)
Now, the time spent in the magnetic field is the distance traveled divided by the velocity. The distance traveled is just half the circumference of the circle, which is πr.
So now we can compute the time spent in the magnetic field:
t = (πr)/v
Substituting the expression for velocity:
t = (πr) / ((mgr)/(qB))
Simplifying:
t = (πr²qB)/mg
Plugging the given values into the equation, we get:
t = (π * (7.7 × 10⁻⁸ kg)² * (11 × 10⁻⁶ C) * (4.2 T)) / (7.7 × 10⁻⁸ kg * 9.8 m/s²)
And after some calculations, the particle spends about 1.66 × 10⁻⁵ seconds in the magnetic field. Quick, but entertaining!
To find the time the particle spends in the magnetic field, we can use the formula for the period of a charged particle in a magnetic field:
T = 2πm / (qB)
Where:
T = period (time spent in the magnetic field)
m = mass of the particle
q = charge of the particle
B = magnetic field strength
Given:
m = 7.7 × 10^(-8) kg
q = +11 μC = 11 × 10^(-6) C
B = 4.2 T
Let's plug in the values and calculate the time spent in the magnetic field:
T = 2π(7.7 × 10^(-8)) / ((11 × 10^(-6))(4.2))
T = 2π(7.7 × 10^(-8)) / ((11 × 10^(-6))(4.2))
T ≈ 1.8 × 10^(-6) seconds
Therefore, the particle spends approximately 1.8 × 10^(-6) seconds in the magnetic field.
To determine the time the particle spends in the magnetic field, we need to calculate its speed and then use the formula for the time taken to complete half a circle.
Step 1: Calculate the speed of the particle.
We can use the formula for the magnetic force on a charged particle moving through a magnetic field:
F = q*v*B,
where F is the magnetic force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field strength.
The magnetic force provides the necessary centripetal force for the particle to travel in a circular path. The centripetal force is given by:
F = mv^2 / r,
where m is the mass of the particle and r is the radius of the circular path.
Setting the magnetic force equal to the centripetal force, we have:
q*v*B = mv^2 / r.
Simplifying the equation, we get:
v = q*B*r / m.
Given values:
q = +11 μC = 11 × 10^-6 C.
B = 4.2 T.
m = 7.7 × 10^-8 kg.
We need to find the radius (r) of the circular path. Since the particle completes half a circle, the radius will be the same as the distance traveled.
Step 2: Calculate the radius of the circular path.
To find the radius of the circular path, we need additional information about the motion of the particle. Specifically, we need either the time taken to complete the half-circle or the distance traveled. In the given question, the distance traveled is not mentioned, so we can't directly calculate the radius.
As a result, we can't determine the exact time spent in the magnetic field without further information.