A wire having a linear mass density of 1.00 g/cm is placed...

Name: Problem 75, Chapter 29: Physics for Scientists and Engineers with Modern Physics
Uploaded: 2020-08-07T08:06:40-08:00
Duration: 8 min 52 s
Channel: Jonathan Ibarra
Description: A wire having a linear mass density of 1.00 g/cm is placed on a horizontal surface that has a coefficient of kinetic friction of 0.200. The wire carries a current of 1.50 A toward the east and slides horizontally to the north at constant velocity. What are (a) the magnitude and (b) the direction of the smallest magnetic field that enables the wire to move in this fashion?

A wire having a linear mass density of 1.00 g/cm is placed on a horizontal surface that has a coefficient of kinetic friction of 0.200. The wire carries a current of 1.50 A toward the east and slides horizontally to the north at constant velocity. What are (a) the magnitude and (b) the direction of the smallest magnetic field that enables the wire to move in this fashion?

Key Concepts

Vector Cross Product and Right-Hand Rule

The cross product between the current direction vector and the magnetic field vector determines both the magnitude and the direction of the force on the conductor. The right-hand rule is a tool used to find the direction of this force. When the right-hand fingers point in the direction of the current and curl toward the magnetic field, the thumb indicates the force direction. This vector operation is foundational in electromagnetism for predicting the behavior of current-carrying conductors in magnetic fields.

Force Equilibrium in Uniform Motion

When an object moves at a constant velocity, it is in a state of dynamic equilibrium, meaning the net force acting on it is zero. In contexts involving both magnetic forces and friction, the magnetic force must exactly balance the frictional force to maintain constant motion. This principle is important as it underpins the analysis and balance of forces in systems where multiple forces interact, allowing for the determination of necessary field magnitudes or other physical parameters.

Magnetic Force on a Current-Carrying Conductor

This concept involves the interaction between a magnetic field and an electric current, which produces a force on the conductor. The force is given by the Lorentz force law, F = I (L × B), where I is the current, L is the length vector of the conductor, and B is the magnetic field. Understanding this principle is crucial when examining how currents can be manipulated by magnetic fields to produce motion or counteract other forces such as friction.

Kinetic Friction

Kinetic friction is the resistive force that opposes the relative motion between two surfaces in contact as one moves over the other. It is characterized by the coefficient of kinetic friction, which, when multiplied by the normal force, yields the magnitude of the frictional force. This concept is important in scenarios where any applied force, such as magnetic force, must overcome friction to maintain constant velocity motion.

Transcript

00:01 So we have a wire from west to east, and we're asked to find the magnetic field magnitude and direction that will cause this wire to go in the north direction, while also having a force, a frictional force that is applied on the surface where the wire stands.

00:26 So we're going to start with part a, where we need to find the magnitude of the magnetic field to cause a velocity to go in this north direction.

00:40 So this velocity is going to be uniform.

00:45 So we're going to write that the force due to the magnetic field is equal to the force due to friction.

01:00 And then we can rewrite this as the magnetic field b times the current i times the length of the wire l is equal to the coefficient of friction times the mass density.

01:22 Times l times g.

01:29 And then the l's will cancel out and we can solve for the magnetic field b and that's going to be equal to mu times row times g divided by i.

01:47 And so we can now find this magnetic field since we know the coefficient of friction which is 0 .2, row is given as 1 gram per centimeter, so 1 gram per centimeter, and then g is given, or we know that g is 9 .8 meters per second squared, and then divided by the given current, which is 1 .5 amps.

02:29 And so this will give us the magnetic field that's equal to 0 .13 tesla's.

02:44 And now to find the direction of this magnetic field, we're going to need to take force components of the force due to the magnetic field.

02:56 So we need to find the y direction, write out the equations for the y direction of the force, and also the x direction of the force.

03:07 So the y direction, the sum of forces, and the y direction, that's going to be equal to the normal force m minus m g.

03:26 Plus f of b times the sine of theta, and this is going to be equal to zero...

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