The microwaves in a microwave oven are produced in a special tube called a magnetron. The electrons orbit in a magnetic field of 9.3×10−2 T at a frequency of 2.6 GHz, and as they do so they emit 2.6 GHz electromagnetic waves.
If the maximum diameter of the electron orbit before the electron hits the wall of the tube is 2.3 cm, what is the maximum velocity the electrons may have?
To find the maximum velocity of the electrons, we can use the relationship between the velocity of a charged particle moving in a circular path and the radius of the circle.
The magnetic field B, given as 9.3×10^(-2) T, is responsible for the centripetal force that keeps the electrons in a circular orbit. The centripetal force is provided by the magnetic force acting on the moving electrons.
The magnetic force F_mag acting on a charged particle with charge q moving with velocity v in a magnetic field B is given by the equation:
F_mag = q * v * B
Since the magnetic force is the centripetal force, we have:
F_mag = (m * v^2) / r
where m is the mass of the electron and r is the radius of the orbit.
Setting the two equations equal to each other and rearranging, we get:
q * v * B = (m * v^2) / r
Simplifying the equation, we can solve for the velocity v:
v = (q * B * r) / m
Given that the radius r is 2.3 cm (or 0.023 m), and the electron charge q is 1.6 × 10^(-19) C, and the electron mass m is 9.1 × 10^(-31) kg, we can substitute these values into the equation:
v = (1.6 × 10^(-19) C * 9.3 × 10^(-2) T * 0.023 m) / (9.1 × 10^(-31) kg)
Calculating this expression, we find the maximum velocity of the electrons as:
v ≈ 1.59 × 10^(-6) m/s
Therefore, the maximum velocity the electrons may have is approximately 1.59 × 10^(-6) m/s.
To find the maximum velocity of the electrons, we can use the equation for the centripetal force experienced by a charged particle moving in a magnetic field:
F = (mv²)/r
Where:
F = Force experienced by the electron
m = Mass of the electron
v = Velocity of the electron
r = Radius of the electron orbit
The force experienced by the electron is due to the magnetic field, given by:
F = qvB
Where:
q = Charge of the electron
v = Velocity of the electron
B = Magnetic field strength
Since the force experienced by the electron is the centripetal force, we can equate the two equations:
qvB = (mv²)/r
Simplifying and solving for v, we get:
v = (qB)r/m
Given:
B = 9.3×10⁻² T (magnetic field strength)
r = 2.3 cm = 0.023 m (radius of the electron orbit)
q = -1.6×10⁻¹⁹ C (charge of the electron)
m = 9.1×10⁻³ⁱ kg (mass of the electron)
Substituting the given values into the equation, we can find the maximum velocity:
v = ((-1.6×10⁻¹⁹ C)(9.3×10⁻² T)(0.023 m))/(9.1×10⁻³¹ kg)
v ≈ -3.72×10⁶ m/s
Note: The negative sign in the velocity indicates that the electrons are negatively charged and move in the opposite direction of the magnetic field.
They make a circle each 1/2.6E9 seconds, so velocity
velocity= distance/time= PI*.023*/2.6E9