Magnetic Force

where q is the charge of the object in question, v is the object's velocity, and B is the applied magnetic field on the object. In other, words the magnetic force is the cross product of the object's velocity and the applied magnetic field multiplied by the object's charge.

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The right-hand rule dictates the directions that the three main variables(the force, velocity, and magnetic field) have to each other.

As you can see, the velocity, force, and field are all perpendicular to one another which means that we have to start applying the rules of three-dimensional space to magnetic fields. So, it's best to introduce the notations behind three-dimensional directions. The first dimension x is just from left to right, the second dimension y is just from vertically down to up, and the third dimension z is from out of the page to into the page. You can think of it as if you were to look at a problem with"3D glasses". If the object is popping out of the page, it will have this notation which means that it is "coming towards you".

The total electromagnetic force acting on a given charge is given through the Lorentz Force Law, which is just the sum of its electric and magnetic forces.

Since magnetic fields are created by currents, or moving charges, we can re-express the magnetic force in terms of current,

which is in terms of the current I, the length of "wire" l, and the magnetic field B. Due to the term with length, this equation usually applies to the magnetic forces generated by wires carrying current.

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To find the direction of the current or the magnetic field, we can use the second right-hand rule to figure it out. The right-hand rule is nothing more than a thumbs-up sign, just like many of us do in real life. But there's more to the thumbs up than a simple gesture. In this second right-hand rule, your thumb points in the direction of the current and your other fingers curl in the direction of the magnetic field. For example, if the current goes straight up, then the magnetic field points into the page(away from you) because your fingers curl away from you. If the current goes into the page, the magnetic field will go downwards.

While it is great to know that magnetic fields are generated by currents, it would be in our best interests to quantify that relationship. Ampere's Law does just that, stating that the magnetic field in space around a given electric current is proportional to that same current itself. This is parallel to how electric fields are created by the charges that serve as their source charges.

Mathematically, Ampere's Law can be stated as

where mu-naught is a constant, known as the permeability of free space, that's equal to 4π×10−7 (tesla-meters)/amperes.

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Ampere's Law is used to find the magnetic field for many different types of magnetic geometries. The following geometries are listed with their given equations for the magnetic field derived from Ampere's Law.

Solenoid Field:

This is a good approximation because we can simplify the integral to a product where we assume the angle is 0 degrees and the magnetic field is constant(which is a safe approximation to make, especially for iron solenoids). Since the solenoid has N loops, we have to multiply that factor into the equation for Ampere's Law and divide out L to get just the magnetic field of the solenoids.

Toroid Field:

This similar to the magnetic field of a solenoid in that we have to multiply over N loops of the toroid. However, the distinction here is that the length of the toroid is really its circumference so that term replaces a standard L term here.

Magnetic field around a long straight wire:

Since the magnetic field of the loop around a long, straight wire is effectively the same as the magnetic field around 1 loop of a toroid, the magnetic field for a current-carrying wire is the same as that of a toroid but without the N term.