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Chapter 27: Problem 30

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

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

Step by step solution

01

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.
02

Formula for the energy of the electron

The energy (E) of an electron is given by the formula: E = (1/2)mv^2
03

Express the velocity in terms of energy

Solving the energy formula for velocity, we get: v = sqrt(2E/m)
04

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)
05

Simplify the expression for radius

Simplifying the expression, we get: r = sqrt(2mE)/(qB)
06

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)
07

Simplify the ratio expression

Canceling the common terms (qB) and (2m) in the numerator and denominator yields: r₁/r₂ = sqrt(E₁/E₂)
08

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))
09

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.

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Most popular questions from this chapter

A particle with charge \(q\) is at rest when a magnetic field is suddenly turned on. The field points in the \(z\) -direction. What is the direction of the net force acting on the charged particle? a) in the \(x\) -direction b) in the \(y\) -direction c) The net force is zero. d) in the \(z\) -direction

A coil consists of 120 circular loops of wire of radius \(4.8 \mathrm{~cm} .\) A current of 0.49 A runs through the coil, which is oriented vertically and is free to rotate about a vertical axis (parallel to the \(z\) -axis). It experiences a uniform horizontal magnetic field in the positive \(x\) -direction. When the coil is oriented parallel to the \(x\) -axis, a force of \(1.2 \mathrm{~N}\) applied to the edge of the coil in the positive \(y\) -direction can keep it from rotating. Calculate the strength of the magnetic field.

An electron (with charge \(-e\) and mass \(m_{\mathrm{e}}\) ) moving in the positive \(x\) -direction enters a velocity selector. The velocity selector consists of crossed electric and magnetic fields: \(\vec{E}\) is directed in the positive \(y\) -direction, and \(\vec{B}\) is directed in the positive \(z\) -direction. For a velocity \(v\) (in the positive \(x\) -direction \(),\) the net force on the electron is zero, and the electron moves straight through the velocity selector. With what velocity will a proton (with charge \(+e\) and mass \(m_{\mathrm{p}}=1836 \mathrm{~m}_{\mathrm{e}}\) ) move straight through the velocity selector? a) \(v\) b) \(-v\) c) \(v / 1836\) d) \(-v / 1836\)

A proton with an initial velocity given by \((1.0 \hat{x}+$$2.0 \hat{y}+3.0 \hat{z})\left(10^{5} \mathrm{~m} / \mathrm{s}\right)\) enters a magnetic field given by \((0.50 \mathrm{~T}) \hat{z}\). Describe the motion of the proton

Which of the following has the largest cyclotron frequency? a) an electron with speed \(v\) in a magnetic field with magnitude \(B\) b) an electron with speed \(2 v\) in a magnetic field with magnitude \(B\) c) an electron with speed \(v / 2\) in a magnetic field with magnitude \(B\) d) an electron with speed \(2 v\) in a magnetic field with magnitude \(B / 2\) e) an electron with speed \(v / 2\) in a magnetic field with magnitude \(2 B\)
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