An electron moves on a circular orbit in a uniform magnetic field. The strength of the field is 1.00×10-5 tesla. The kinetic energy of the electron is 178.0 eV. (1 eV = 1 electron volt = 1.6 ×10−19 joules.) Calculate the radius of the orbit.
(electron parameters: the charge is -e where e = 1.602×10-19 C; the mass is m = 9.11×10-31 kg.)
KE =m•v²/2,
v= sqrt(2•Ke/m) = sqrt(2•178•1.6•10^-19/9.11•10^-31) =7.9•10^6.
Lorentz force
F=q•v•B•sinα,
Since the circular orbit, sinα = 1.
mv²/R = q•v•B.
R= m•v/q•B = m•v/e•B =
=9.11•10^-31•7.9•10^6/1.6•10^-19•1•10^5 =4.5•10^-10 m
To calculate the radius of the orbit of an electron moving in a uniform magnetic field, you can use the equation for the centripetal force. The centripetal force acting on a charged particle moving in a circular path in a magnetic field is given by:
F = (q * v * B)
Where F is the centripetal force, q is the charge of the electron, v is the velocity of the electron, and B is the magnetic field strength.
In this case, we need to equate this force to the force required to sustain the circular motion of the electron due to its kinetic energy. The kinetic energy is given by:
K.E. = (1/2) * (m * v^2)
Where K.E. is the kinetic energy, m is the mass of the electron, and v is its velocity.
Now, let's calculate the velocity of the electron using the given kinetic energy.
Given:
K.E = 178.0 eV = 178.0 * (1.6 x 10^-19) J (converting eV to joules)
Now, equating the kinetic energy with the energy of motion equation:
(1/2) * (m * v^2) = 178.0 * (1.6 x 10^-19) J
Rearranging the equation:
v^2 = (2 * 178.0 * (1.6 x 10^-19) J) / m
v^2 = (2 * 178.0 * (1.6 x 10^-19) J) / (9.11 x 10^-31) kg (substituting the given mass of the electron)
v^2 ≈ 6.588 x 10^23 m^2/s^2
Taking the square root of both sides to find the velocity:
v ≈ sqrt(6.588 x 10^23) ≈ 2.57 x 10^11 m/s
Now we can substitute this velocity into the equation for the centripetal force:
F = (q * v * B)
F = (-e * v * B) (since the charge of the electron is -e)
We know that the force required for circular motion is equal to the centripetal force:
F = (m * v^2) / r
Substituting the values we have:
(-e * v * B) = (m * v^2) / r
(-e * v * B) = (m * v^2) / r
(-e * v * B * r) = (m * v^2)
Rearranging the equation to solve for the radius:
r = (m * v^2) / (-e * v * B)
Substituting the given values:
r = (9.11 x 10^-31 kg * (2.57 x 10^11 m/s)^2) / (1.602 x 10^-19 C * 1.00 x 10^-5 T)
Simplifying this equation will give us the value of the radius of the electron's orbit.