Two electrons move in a uniform magnetic field. Their speeds and entry directions are varied to compare the radii of their paths and the time each takes to complete 1 revolution.
Adjust conditions:
The sliders adjust the values
同じ磁場に垂直に入る電子の速さを2倍にすると、1周にかかる時間は何倍か？
磁場に垂直な速度成分を保ち、平行成分を加えて速さを2倍にすると半径は？
The equations explain the phenomenon
1The force on a moving electric charge is expressed as follows.F=q v × B……
2The dot product of each side of the equation of motion with the velocity reveals whether the speed changes.d(½ m v²)dt=F · v=0……
3The magnetic force and centripetal acceleration determine the radius of the circle.m v²r=|q| v B⇒r=m v|q| B……
4Dividing the circumference by the speed gives the time for 1 revolution.T=2π m|q| B……
5The velocity is resolved into components parallel and perpendicular to the magnetic field.r=m v⊥|q| B，p=v∥ T……
Extension: Electric Currents and Magnetic Fields
6The result applies to electrons in a wire.F=(n S L) · e v B=I L B……
7The magnetic field produced by an electric current is expressed as follows.B=μ₀ I2π r……
8The magnetic fields at the center of a circular current loop and inside a long solenoid are calculated.B=μ₀ n I……
9The force between 2 parallel wires is calculated.FL=μ₀ I₁ I₂2π d……
10Rotating the coilN=I S B sinθ……
What this chapter has shown
An electron entering a uniform magnetic field perpendicularly experiences a force −e v×B perpendicular to its velocity. This force does no work, so the speed remains constant. It provides the centripetal force, causing the electron to trace a circle of radius mv/(eB). The time for 1 revolution, 2πm/(eB), is independent of speed.
1.000 timesPeriod ratio when the speed is doubled
Adding a parallel velocity component while keeping the perpendicular component unchanged in the same magnetic field produces helical motion with the same radius and period.
In everyday lifeA cyclotron uses the independence of the time for 1 revolution from speed. A magnetic field bends the paths of particles such as protons, while an electric field accelerates them repeatedly.
A magnetic field exerts a force only on moving electric charges, perpendicular to their velocity. This force does no work, so a charged particle's speed remains constant as it moves in a circle, and its period is independent of its speed. The velocity component parallel to the magnetic field is unaffected, so a charged particle entering the field at an oblique angle follows a helical path. The force on a wire is the sum of the forces on the electrons inside it. Both the forces between electric currents and the operation of motors arise from the same Lorentz force.