Unlike other interactions, the forces between two moving charges have certain peculiarities: they are not equal in module and do not have the same direction. Be careful, do not attemp to make analogies with forces from other sources that you have studied.

Magnetic force

Consider a punctual charge (blue ball) with a speed \(\vec{v}\) (blue vector). The magnetic force \(\vec{F_m}\) is orthogonal to the (yellow) plane where the charge velocity vector \(\vec{v}\) and the magnetic field vector \(\vec{B}\) (green vector) are contained.
Definitions:
Current Element
Moving electrons form an electrical current. We call current element , the amount of electric charges that crosses orthogonally some portion of the conductor, at some time interval. The direction of this vector is the same of charges displacement. The electric current intensity unit in the \(IS\) is Ampere \((A = \frac{C}{s})\) .
Magnetic Permeability \((\mu)\)
Different mediums will affect the magnet fiels in a different maner, the magnetic permeability is used to account for this when calculating the force acting on a current element in the presence of a magnetic field in a specific medium. The permeability of vacuum (absence of material medium) represented by \(\mu_0\) , it: $$\mu_0 = 4 \pi 10^{-7} \frac{Tm}{A}$$ noting that \(\pi\)is worth approximately 3.14.

Lorentz Force

It is the magnetic force \(\vec{F_m}\) which acts on the electrified particle \(q\) when it moves with velocity \(\vec{v}\) in the region of a magnetic field \(B\) . This force has the following characteristics:

Direction
Perpendicular to the plane determined by \(\vec{v}\) and \(\vec{B}\), and it is given by the "right hand rule". When the charge \(q\) is negative, the direction is inverted.
Intensity
It is given by: $$F_m = qv B ~sen(\theta)$$ Where \(\theta\) is the angle between the vector \(\vec{v}\) and \(\vec{B}\) .
It is possible to write the Lorentz force more concisely using a vector product: $$\vec{F_m} = q \vec{v} \times \vec{B}$$

Applications of Magnetic Force

Loads in a Magnetic Field
A moving charge in a uniform magnetic field experiences a force that is perpendicular to speed. The trajectory of this charge is a circle and the centripetal force is the magnetic force itself. In the case of a charged particle of mass \(m\) and absolute charge \(|q|\) , moving with velocity \(\vec{v}\) perpendicular to a uniform magnetic field \(B\) , the radius of the circular path is: $$r = \frac{mv }{ q B}$$
Linear Conductor in a Magnetic Field
For a linear conductor of length \(L\) , traversed by a current of intensity \(i\) , immersed in a uniform magnetic field \(\vec{B}\) with \(\theta\) being the angle between the conductor and \(\vec{B}\) , the magnetic force is given by: $$F_m = i BL ~ sen(\theta)$$
Force Between Conductors
Consider a long wire, traversed by an electric current. Consider an eletric current element that is parallel to the wire. It is found experimentally that the eletric current element is subject to the action of a force that has the following characteristics:
  • There is an attractive force when the directions of the two currents, flowing through the wire and the current element, have the same direction. The force is repulsive if the direction of one current is in the opposite direction of the other;
  • The strength of the force is directly proportional to the intensity of the current flowing through the wire, by the current element, and inversely proportional to the distance that separates them.
In quantitative terms, for two conductors, parallel and straight, with length \(L\) and traversed by eletric currents \(i_1\) and \(i_2\) , separated by a distance \(d\), they interact with a magnetic force of intensity: $$F_m = \frac{\mu_0}{2\pi} \frac{i_1 i_2 L}{d}$$

Rule of the Right Hand

To find the direction of the magnetic force, it is common to adopt the Rule of the right hand, where one uses the fingers of his right hand, as illustrated in the figure, to find the direction of the vectors. The index finger must point in the direction \(\vec{v}\) and the middle finger in the direction of \(\vec{B}\), so the thumb points in the direction of the magnetic force, \(\vec{F}_m\) .

Hall Effect

When a conductive sheet of thickness \(l\) is traversed by a eletrical current \(i\), and is subjected to a magnetic field \(\vec{B}\) perpendicular to it, some charge carriers \(q\) accumulate on one side of the sheet, due to the magnetic force, creating a potential difference \(\mathbb{V}\) between the sides of the sheet. From the polarity of the sides one can find the sign of the charge carriers.