The equations and how to use them

Coulomb's Law:

Used to calculate the force between two charges The units of force are the Newton Dielectric constant of free space = 8.85 * 10^-12 It is important to remember force is a vector quantity

Electric Fields:

This equation is used to calculate the electric field generated by a charge Electric field strength is measured in volts per metre The sign of E depends on the sign of Q Electric field strength is a vector quantity

This equation is the result of combining Coulomb's law and the equation for electric field strength

Potential Energy:

This equation is the work done moving a charge a distance This can be used to define what voltage is:

"The work done to move a unit charge between the two points is equal to the potential difference between those two points"

Note: Voltage is a scalar quantity

This equation shows that the electric field strength of a charge is equal but opposite to voltage differentiate with respect to x (distance between the charge and the reference point).

Gauss' Law:

"The electric flux passing through any closed surface is equal to the total charge contained within that surface."

The above equation demonstrates this law in action When the Gaussian surface is irregular you need to break the surface into small areas and calculate the electric field normal to the surface and then integrate over the whole surface and multiply by dielectric constants

This is the dot product, which is required for irregular surfaces and simply refers to taking the component of the electric field that is normal to the surface. This produces a scalar quantity.

Electrical Displacement and Polarisation:

This shows the origins of relative permittivity, as it causes the induced charges to set up an opposing electric field. This leads to the "original field weakening"

This is simply a rearrangement of the above equation to have polarisation as the subject

Capacitance:

This is derived as voltage between two conductors is proportional to the charge transferred between them. In this case capacitance is the constant of proportionality for the relationship.

This is the equation for capacitance between two parallel calculated by first finding the equations for Q and V This method of finding capacitance can be used for a variety of different objects

Need to review how to calculate the potential difference between a variety of surfaces

Electrostatic Energy:

The energy stored stored by the collection of charge can be derived as follows:

This starts by considering an isolated body with charge Q and then adding small amounts of charge δQ

We can also calculate energy density this is done by first rewriting the equation for energy stored changing the capacitance to its equation as follows:

Force on a current carrying conductor in a uniform magnetic field

Where: B is magnetic flux density I is current flowing in the conductor L is length of the conductor ⍺ is the angle between the field normal to the force and the wire

Current flow

Where: q = Charge of a single electron/charge carrier n = number of electrons/charge carriers A = cross-sectional area of wire v = velocity of charge carriers

Resistivity

This equation is used then used to derive the most general form of Ohm's law:

Where J is the current density

Magnetic Field Strength

We know that magnetic flux density is:

Where H is as follows:

The angle θ is the angle between H and the tangent to the loop around the wire.

For a circle H is tangential to the path therefore giving the following equation:

Biot-Savart Law

This is very similar to separating up of a charge as done in electrostatics, it states that:

This is used for anything other than a straight wire of infinite length

Faraday's Law

States that whenever the flux through a loop changes with time, then there's a voltage induced in the loop. This voltage is typically referred to as the electromotive force. The law can be demonstrated through the following equation:

The negative signifies that the emf opposes the change that caused it Since Φ is total flux linkage increasing the number of turns increases the emf

This equation can be used to derive the equation for a perfect transformer as the N is taken outside of the total flux linkage equation.

Self and Mutual Induction

In a solenoid the magnetic field strength H = n * I If the solenoid is air filled then B = μo * n * I Using this equation to find the emf induced in a solenoid produces the following equation:

Comparing this equation with the equation self-induced in an inductor, as shown below:

Allows the derivation of an equation for induction (L):

For mutual induction in linear circuits the mutual induction between the two is exactly the same and is calculated using the following equation:

Energy Stored by an Inductor

As power is the product of voltage and current it is very easy to derive an equation of energy stored:

As power is rate of change of energy this means energy stored in an inductor is:

The above equation holds true even for DC as there is always energy stored in a magnetic field