A long, straight wire carries a 13.0 A current. An electron...

A long, straight wire carries a 13.0 A current. An electron is fired parallel to this wire with a velocity of $280 \mathrm{~km} / \mathrm{s}$ in the same direction as the current, $1.80 \mathrm{~cm}$ from the wire. (a) Find the magnitude and direction of the electron's initial acceleration. (b) What should be the magnitude and direction of a uniform electric field that will allow the electron to continue to travel parallel to the wire? (c) Is it necessary to include the effects of gravity? Justify your answer.

Key Concepts

Magnetic Field Around a Current-Carrying Conductor

A long, straight current?carrying wire produces a magnetic field in the space around it that can be determined using Ampère’s law. The field circulates around the wire and its magnitude at a given radial distance is given by B = (??I)/(2?r), where I is the current and r is the distance from the wire. This concept is fundamental in electromagnetism, as it describes how moving charges (currents) generate magnetic fields and how these fields diminish with distance.

Lorentz Force

The Lorentz force law describes the force exerted on a charged particle moving within electric and magnetic fields. For a magnetic field, the force is given by F = q(v × B), meaning that the force is perpendicular to both the velocity of the particle and the magnetic field. This force is responsible for the acceleration of charged particles when they move in proximity to current-carrying conductors and is central to understanding particle trajectories in magnetic fields.

Electric Force and Field Cancellation

In contexts where a charged particle is subject to a magnetic force that would cause deflection, a uniform electric field can be applied to counteract this force. The electric force acting on a charge is F = qE, and by carefully choosing the magnitude and direction of the electric field, the net electromagnetic force can be set to zero, allowing the particle to continue its motion undisturbed. This principle is widely used in charged particle beam control and in devices that require precise steering of electron paths.

Newton’s Second Law in Charged Particle Dynamics

Newton’s second law, F = ma, connects the forces acting on a particle to its acceleration and mass. When applied to charged particles in electromagnetic fields, this law allows us to determine how the magnetic and electric forces affect the motion of the particle. Understanding this relationship is essential for predicting motion, calculating acceleration, and analyzing trajectories in fields generated by currents and applied potentials.

Negligibility of Gravitational Force in Electromagnetic Contexts

In many electromagnetic scenarios, especially those involving small particles like electrons, the gravitational force is several orders of magnitude weaker than the electromagnetic forces. This concept emphasizes that, for electrons and similar charged particles, the effects of gravity can often be ignored in calculations as they do not significantly influence the particle’s trajectory when compared to the dominant electromagnetic forces.

Transcript

00:01 Hi everyone, this is the problem based on magnetic effects of current and force on the moving charge through the magnetic field.

00:09 Here it is given in a very long straight conductor, current of 13 ampere is flowing and electron is projected parallel to it with the speed of 250 km per second or 2 .5 into 10 to the power.

00:35 5 meter per second from a distance at a distance 2 cm from it that is 0 .02 meter we have to find the magnitude of acceleration we have to find direction and magnitude of electric fields so that electron remains undeviated from its paths und deviated from its paths in c part is it necessary to neglect gravity effect.

01:51 Let us see here.

01:53 Magnetic field at the position of projection of electron will be mu not i upon 2 pi r and due to this magnetic field force on the electron will be few b sign of theta.

02:10 Magnetic field will be perpendicular to plane of projection of electron.

02:16 You can see the direction here.

02:17 It will be this will be the direction charge is e magnetic field is mu not i upon two pi speed into cyan of 90 and this force will be skull to mass into acceleration so you will get mu not i e b upon two pi r so magnitude of acceleration you will get mu not i e b upon two pi m or substitute the value mu naught is 4 pi into 10 to the power minus 7 current is 13 ampere charge on electron 1 .6 into 10 to the power minus 19.

03:04 Speed of electron is 2 .5 upon 10 to the power 5.

03:09 Mass of electron 9 .11 10 to the power minus 31 and radius is two distances to centimeter that is 0 .02...

Please give Ace some feedback