5/7 Magnetic field and forces

Todays class began with the discussion of magnetic fields and continued with magnetic forces. A brief definition of our study is as follows: A magnetic field is the magnetic effect of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude.  A magnetic force is the attraction or repulsion that is created between electrically charged particles because of their motion.  

We began class with the discussion of magnetic fields, here we have a bar magnetic and around it are magnetic field lines. The magnetic field direction is defined as the direction in which the dark end or north pole of a compass needle is oriented in the presence of the field of interest. As is pictured below:

Activity:  Field Directions Around a Bar Magnet

a. Use a small compass to map out the magnetic field surrounding a large bar magnet;  denote the direction of the field with arrows.  Don’t forget to include the region “in” the magnet between the north and south poles.  Sketch lines with arrows in the space below.

b. Take a look at a bar magnet with iron filings sprinkled around it. Draw a series of magnetic field lines “in” and around the magnet.  Assume the lines are continuous across the boundaries between the magnetic material and the surrounding air.  Mark lines with directional arrows pointing in the direction of the north pole of your compass.

c. Pretend you are in a two-dimensional world.  (Flatland again!) Draw several closed loops in the space in part b.  Let one loop enclose no magnetic pole, another loop enclose one of the poles, and another enclose both poles.  Assuming that each line coming into a loop is negative and each line coming out is positive, what is the net number of magnetic field lines coming in and out of a loop in each case

Part C is pictured below.

d. Now we come to a magnetic equivalent to Gauss’s law describing the net magnetic flux, Φ_m, coming out of a closed three-dimensional surface.  Can you guess what Φ_m is equal to?  

We found that flux magnetic field was equal to

Gauss's Law for Magnetism

The magnetic flux into a surface is equal to the magnetic flux out of a surface. 

Flux Through a Loop 

When both the direction and magnitude of the magnetic field does not change across the surface bounded by a closed loop, the magnetic flux can be expressed as: 

In the picture below we had a discussion of how force and velocity is oriented depending in which direction magnetic field is pointing. (Upper left hand corner) If magnetic field is pointing down (north pole) and velocity pointing out for the top (pictured) then the force vector is perpendicular (pointing to the left). (Top right-hand corner) here we have the magnetic field pointing up (south pole) with the velocity pointing out, then force vector is again perpendicular to both. The bottom two scenarios are similar to the two on top and what we concluded was that the force vector is perpendicular to the magnetic field and velocity vectors.

Activity:  The Magnetic Force Exerted on Moving Charges—An Electron Beam in an Oscilloscope

Here we have our next experiment, where Professor Mason moved the north pole of the magnet parallel and then perpendicular to the electron beam in an oscilloscope. When he put it parallel nothing happened, but when he put it perpendicular to the oscilloscope it caused the electron beam to move. This is due to magnetism. Magnetism is the force that moving charges exert on one another. Therefore, when an magnetic field was introduced it caused the electron to move perpendicular to the field.

Electricity is the force that charges exert on one another. Since this force exists whether or not the charges are moving, it is sometimes called the electrostatic force. The combination of electric and magnetic forces on a charged object is known as Lorentz force.

Magnetic Force on a Moving Charge: 

The force on a moving charge in a magnetic field is equal to the cross product of the particles velocity with the magnetic field times the magnitude of the charge. 

Activity:  Using the Lorentz Force in Calculations

a. Consider a proton traveling at θ = 30 degrees with respect to a magnetic field of strength 2.6 ◊ 10^-3 T as shown .  It has a speed of 3.0 ◊ 10⁶ m/s.  What is the magnitude and direction of the force exerted on the proton by the magnetic field?

b. If the particle is an electron instead, what is the magnitude and direction of the force exerted on the particle by the magnetic field?

Our answers to A and B

 Free Charge Moving in Uniform Magnetic Field

The general path of a moving charge in a constant magnetic field is that of a helix with its axis parallel to the direction of the magnetic field.

The component of velocity of the charged particle that is parallel to the magnetic field is unaffected, i.e. the charge moves at a constant speed along the direction of the magnetic field.

If the particle has a component of velocity parallel to the magnetic field, then its circlular motion will drift at a constant speed (equal to that of its parallel-velocity component, v_ll) along the magnetic field producing an overall helical motion. 

The component of the velocity perpendicular to the magnetic field, is what cause the particle to executes uniform circular motion perpendicular to the magnetic field.

The radius of the circle can be found by equating the magnetic force on the charge with the centripetal force.

Here we Calculated angular frequency with what we learned in free charge moving in a uniform  magnetic field.

MAGNETIC FORCE ON CURRENT-CARRYING WIRE

Here we derived the formula for magnetic force. We used the definition of current, velocity and Lorentz Force to create the equation needed in order to perform such calculations.

Magnetic Force on a  Current Loop
Here we have a rectangular loop that carries current in the presence of a magnetic field. We drew the direction of the forces, magnetic field and discussed torque. We found that the loop would turn 90 degrees then stop when current was applied.

 

Magnetic Force on a Current Carrying Loop Problem 

Pictured below is a magnetic field in 15 segments, for homework we were asked to create an excel spread sheet that included force, radius, magnetic field, the angle and current (pictured below). To summarize, we found that the force was stronger as the angle become more perpendicular.