An electron is accelerated from rest through potential...

An electron is accelerated from rest through potential difference $V$ and then enters a region of uniform magnetic field, where it undergoes uniform circular motion. Figure $28-38$ gives the radius $r$ of that motion versus $V^{1 / 2} .$ The vertical axis scale is set by $r_{s}=3.0 \mathrm{~mm}$, and the horizontal axis scale is set by $V_{s}^{1 / 2}=40.0 \mathrm{~V}^{1 / 2} .$ What is the magnitude of the magnetic field?

An electron is accelerated from rest through potential difference $V$ and then enters a region of uniform magnetic field, where it
undergoes uniform circular motion. Figure $28-38$ gives the radius $r$ of that motion versus $V^{1 / 2} .$ The vertical axis scale is set by $r_{s}=3.0 \mathrm{~mm}$, and the horizontal axis scale is set by $V_{s}^{1 / 2}=40.0 \mathrm{~V}^{1 / 2} .$ What is the magnitude of the magnetic field?

Key Concepts

Energy-Velocity Relationship

When a charged particle is accelerated from rest through a potential difference, the work done by the electric field (equal to the product of the charge and the potential difference) is converted into kinetic energy. This connection allows us to relate the particle's speed to the accelerating voltage via the equation (1/2)mv² = qV, where m is the mass and q is the charge of the particle.

Motion in a Uniform Magnetic Field

A charged particle entering a region of uniform magnetic field experiences the Lorentz force, which acts perpendicular to its velocity. This force does not work on the particle but causes it to undergo uniform circular motion. The radius of the orbit is given by r = mv/(qB), where B represents the magnetic field strength.

Combining Energy and Circular Motion

By substituting the speed derived from the energy-velocity relationship into the radius expression for circular motion, one obtains a formula that shows the radius is proportional to the square root of the accelerating potential. Specifically, r = ?(2mV/(q))/B, emphasizing how changes in the potential difference affect the curvature of the particle's path.

Graphical Analysis and Slope Interpretation

Plotting the radius of circular motion versus the square root of the accelerating potential yields a straight line. The slope of this line is directly related to the physical constants, including the mass, charge, and the magnetic field strength. By analyzing this linear relation, one can extract the magnitude of the magnetic field from the experimental or graphical data.

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