What magnetic field is required to constrain an electron with a kinetic energy of 400eV to a circular path of radius .80m? (answer: 8.4x10^-5T) What equation(s) do I use? Any help is appreciated!
V = sqrt (2E/m) relates velocity to energy.
E = 400 eV *1.6*10-19 J/eV = 6.4*10^-17 J
e V B = m V^2/R
That equates the magnetic force to the centripetal force
B = m V/(R e) = [sqrt(2 E m)]/(R e)
To find the magnetic field required to constrain an electron to a circular path, you can use the following equation:
B = (mv) / (eR)
where:
- B is the magnetic field
- m is the mass of the electron (9.10938356 × 10^-31 kg)
- v is the velocity of the electron
- e is the charge of the electron (-1.602176634 × 10^-19 C)
- R is the radius of the circular path
Since the kinetic energy is given, we can calculate the velocity (v) using the formula:
K.E. = (1/2)mv^2
rearranging the equation:
v = sqrt((2 * K.E.) / m)
Substituting the given values into the equations:
K.E. = 400 eV = 400 * (1.602176634 × 10^-19 J)
m = 9.10938356 × 10^-31 kg
R = 0.8 m
v = sqrt((2 * 400 * (1.602176634 × 10^-19 J)) / (9.10938356 × 10^-31 kg))
v ≈ 2.1872 × 10^6 m/s
Now, we can substitute the calculated values into the magnetic field equation:
B = ((9.10938356 × 10^-31 kg) * (2.1872 × 10^6 m/s)) / ((-1.602176634 × 10^-19 C) * (0.8 m))
B ≈ 8.4 × 10^-5 T
Therefore, the magnetic field required to constrain an electron with a kinetic energy of 400eV to a circular path of radius 0.80 m is approximately 8.4 × 10^-5 Tesla (T).
To find the magnetic field required to constrain an electron to a circular path, you can use the equation for the Lorentz force.
The Lorentz force (F) acting on a charged particle moving in a magnetic field is given by the equation:
F = q * v * B
Where:
- F is the force applied to the particle,
- q is the charge of the particle,
- v is the velocity of the particle, and
- B is the magnetic field strength.
In this case, the force acting on the electron is the centripetal force, given by:
F = (m * v^2) / r
Where:
- m is the mass of the electron,
- v is the velocity of the electron, and
- r is the radius of the circular path.
Since the electron has kinetic energy, you can find its velocity using the equation:
K.E. = (1/2) * m * v^2
Rearranging this equation to solve for v:
v = sqrt((2 * K.E.) / m)
Substituting this expression for v into the equation for the Lorentz force:
(m * v^2) / r = q * v * B
Rearranging this equation to solve for B:
B = (m * v) / (q * r)
Finally, substitute the given values into the equation to get the magnetic field:
B = (m * v) / (q * r) = (9.10938356 × 10^-31 kg * sqrt((2 * 400 eV * 1.602176634 × 10^-19 J/eV) / (9.10938356 × 10^-31 kg))) / ((1.602176634 × 10^-19 C) * 0.80 m)
Evaluating this expression will give you the magnetic field required to constrain the electron to a circular path.