Question

An electron with energy equal to \(4.00 \cdot 10^{2} \mathrm{eV}\) and an electron with energy equal to \(2.00 \cdot 10^{2} \mathrm{eV}\) are trapped in a uniform magnetic field and move in circular paths in a plane perpendicular to the magnetic field. What is the ratio of the radii of their orbits?

Short answer

Answer: The ratio of the radii of the orbits of the two electrons is √2.

Step-by-step solution

Formula for the radius of the circular orbit

The formula for the radius (r) of the circular orbit an electron with charge (q) and velocity (v) moving in a plane perpendicular to a magnetic field (B) is given by: r = mv/(qB), where m is the mass of the electron.

Formula for the energy of the electron

The energy (E) of an electron is given by the formula: E = (1/2)mv^2

Express the velocity in terms of energy

Solving the energy formula for velocity, we get: v = sqrt(2E/m)

Substitute the velocity expression back into the radius formula

Substituting the expression for velocity into the equation for the radius, we get: r = m(sqrt(2E/m))/(qB)

Simplify the expression for radius

Simplifying the expression, we get: r = sqrt(2mE)/(qB)

Ratio of the radii of the electrons' orbits

Let r₁ and r₂ be the radii of the orbits for the electrons with energies E₁ and E₂ respectively. The ratio of the radii is given by: r₁/r₂ = sqrt(2mE₁)/(qB) * (qB)/sqrt(2mE₂)

Simplify the ratio expression

Canceling the common terms (qB) and (2m) in the numerator and denominator yields: r₁/r₂ = sqrt(E₁/E₂)

Calculate the ratio

Given the energies E₁ = 4.00*10² eV and E₂ = 2.00*10² eV, we can now calculate the ratio of the radii: r₁/r₂ = sqrt((4.00*10²)/(2.00*10²))

Simplify the result

Simplifying the result, we get: r₁/r₂ = sqrt(2) So the ratio of the radii of the orbits of the two electrons is √2.