Magnetic Forces, Fields, & Current

In the image below, Mason showed us what is looks like when iron shavings are sprinkled on top of a magnet.

Mason showed us that a magnet placed near the oscilloscope will cause the visual dot on the screen to move away from the magnet. 

The next 3 images are of a charged wire subjected to a magnetic field. When the wire was charged it jumped forward.

Magnetic flux measures the number of magnetic field lines that pass perpendicularly through a surface.  If you cut the domain of a magnet in half, there will be zero magnetism.

At every point in space surrounding a magnet, a moving charge carrying current creates a force field. This force field has both magnitude and direction. 

Gauss' Law for Magnetism states that the magnetic flux into a surface is equal to the magnetic flux outside of a surface.

 Here we solved for the acceleration of an electron moving through an magnetic field.

We then solved a different problem that asked us to find the magnetic field when given frequency. The formulas are shown.

Here we drew the field lines around a magnet. The magnet was in the center of the field lines. The field lines leave the north pole and enter the south pole. 

The formula for finding flux through a loop is as follows:

The magnetic force on a moving charge is found using these formulas. The direction of the vectors is shown as well. The direction of the magnetic force is always at a right angle to the plane formed by the velocity vector and the magnetic field. The magnetic force depends upon the size of the charge, the magnitude of the electric field, and the velocity of the charged particle moving perpendicular to the magnetic field. When the velocity vector is parallel to the vector of the magnetic field, then the force is zero. When the charge is negative, the force will be in the opposite direction but will have the same magnitude. 

Since the magnetic force always acts at right angles to the velocity vector, the magnetic force cannot do work on the charge. The magnetic field does not change the velocity of a charge, only its direction. The velocity vector component parallel to the magnetic field is unaffected. Negatively charged particles circulate in opposite directions of the positively charged particles. The formulas for angular velocity, the parallel component of velocity and the perpendicular component of velocity are as follows:

Below are the formulas for finding the magnetic forces on a current carrying wire.

Here is a derivation for the magnetic force on a current carrying conductor:

The loop shown below demonstrates the direction of current on a current carrying wire.

The drawings show the direction the currents, magnetic fields and magnetic forces. On the left, the summation of magnetic forces is equal to zero because the forces at the top and bottom of the loop are parallel to the magnetic field and the forces on the left and right side of the loop have the same magnitude but opposite directions. The torque at the top of the loop is equal to that of the bottom of the loop.

Here is a depiction of the force field in a 3D plane. We can see that they forces cancel out.

Magnetic forces on a semicircular wire can be found using the following derivations.

In the excel spread sheet below, I divided the semicircle into 20 sections and found the magnetic force at each section using the given magnetic field, current and radius and found the net force.

Summary:

• Magnetic flux measures the number of magnetic field lines that pass perpendicularly through a surface.  
• If you cut the domain of a magnet in half, there will be zero magnetism. 
• At every point in space surrounding a magnet, a moving charge carrying current creates a force field. 
• This force field has both magnitude and direction. 
• Gauss' Law for Magnetism states that the magnetic flux into a surface is equal to the magnetic flux outside of a surface. 
• The magnetic force on a moving charge is found using these formulas. The direction of the vectors is shown as well. 
• The direction of the magnetic force is always at a right angle to the plane formed by the velocity vector and the magnetic field. 
• The magnetic force depends upon the size of the charge, the magnitude of the electric field, and the velocity of the charged particle moving perpendicular to the magnetic field. 
• When the velocity vector is parallel to the vector of the magnetic field, then the force is zero. 
• When the charge is negative, the force will be in the opposite direction but will have the same magnitude.