ssm At a certain location, the horizontal component of the earth's magnetic field is 2.5 ×10^-5

Ssm At a certain location, the horizontal component of the...

Key Concepts

Lorentz Force

The Lorentz force describes the force exerted on a charged particle moving through a magnetic field. It is given by the equation F = q(v × B), where q is the charge, v is the velocity vector, and B is the magnetic field vector. This force acts perpendicular to both the velocity and the magnetic field, and its magnitude is qvB sin?, with ? being the angle between the velocity and magnetic field.

Magnetic Field

A magnetic field is a vector field that exerts forces on moving electric charges and magnetic dipoles. In the context of the Earth, the horizontal component of the Earth's magnetic field affects charged particles moving near the Earth's surface, and its direction and magnitude determine the resultant magnetic force experienced by those particles.

Gravitational Force

Gravitational force is the attractive force between two masses, most notably the weight that an object experiences due to the gravitational field of the Earth. It is given by F = mg, where m is the mass of the object and g is the acceleration due to gravity.

Force Equilibrium (Balance of Forces)

Force equilibrium occurs when all forces acting on an object balance each other, resulting in no net force and thus no acceleration. In problems like this, equilibrium is achieved by setting the magnetic force equal to the gravitational force, allowing one to solve for unknown quantities such as the speed of the charged particle.

Right-Hand Rule

The right-hand rule is a mnemonic for determining the direction of the force experienced by a moving charge in a magnetic field. It involves orienting your fingers in the direction of the velocity, curling them towards the magnetic field direction, with your thumb then indicating the direction of the resulting magnetic force for a positive charge.

Transcript

00:01 Okay, let's begin this problem by looking at a top -down view of the ground, or at least a rectangular or portion of it.

00:12 In this view, directly up will be towards the north, directly down towards the south.

00:18 To the right would be east, and to the left would be west.

00:24 Now, we're given that the horizontal component of the magnetic field is headed directly to the north, and we know that a proton, which we represent here as a dot with charge plus e, is headed directly to the east at some velocity v.

00:47 We're given that the speed of this proton is such that the magnetic force that the proton feels is equal and opposite to the weight that the proton feels due to gravity.

01:01 If we use the right -hand rule, we see that the magnetic force is directed out of the screen or directly up and we'll represent that as a dot with a circle around it and of course the gravitational force is going to be headed directly downward because it's going to go towards the ground and that is represented as a vector with a it's an x with a circle around it now because we know that they're balanced we know that the magnitudes of these forces are also balanced so we can say that the magnitude of the magnetic force is equal to the magnitude of the gravitational force, of course, acting on this project.

01:48 From the textbook, the magnitude of the magnetic force is given by the charge of the particle, multiplied by its speed, multiplied by the magnitude of the magnetic field through which the particle is traveling, and then lastly, multiplied by the sign of the angle between the magnetic field vector and the velocity vector of the object we care about.

02:08 Be equal to the gravitational force of this on this object and that is just the mass of that object or the proton multiplied by the acceleration due to gravity.

02:16 The question asked what speed allows for this balance? that means we want to solve for the variable v and we're going to do that now...

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