Ohm’s Law
For many conductors of electricity, the electric current which will flow through them is directly proportional to the voltage applied to them. When a microscopic view of Ohm’s law is taken, it is found to depend upon the fact that the drift velocity of charges through the material is proportional to the electric field in the conductor. The ratio of voltage to current is called the resistance, and if the ratio is constant over a wide range of voltages, the material is said to be an “ohmic” material. If the material can be characterized by such a resistance, then the current can be predicted from the relationship:
// Data can be entered into any of the boxes below. Specifying any two of the quantities determines the third. After you have entered values for two, click on the text representing to third in the active illustration above to calculate its value.
Voltage Law
The voltage changes around any closed loop must sum to zero. No matter what path you take through an electric circuit, if you return to your starting point you must measure the same voltage, constraining the net change around the loop to be zero. Since voltage is electric potential energy per unit charge, the voltage law can be seen to be a consequence of conservation of energy.
The voltage law has great practical utility in the analysis of electric circuits. It is used in conjunction with the current law in many circuit analysis tasks.
The voltage law is one of the main tools for the analysis of electric circuits, along with Ohm’s Law, the current law and the power relationship. Applying the voltage law to the above circuits along with Ohm’s law and the rules for combining resistors gives the numbers shown below. The determining of the voltages and currents associated with a particular circuit along with the power allows you to completely describe the electrical state of a direct current circuit.
Current Law
The electric current in amperes which flows into any junction in an electric circuit is equal to the current which flows out. This can be seen to be just a statement of conservation of charge. Since you do not lose any charge during the flow process around the circuit, the total current in any crosssection of the circuit is the same. Along with the voltage law, this law is a powerful tool for the analysis of electric circuits.
The current law is one of the main tools for the analysis of electric circuits, along with Ohm’s Law, the voltage law and the power relationship. Applying the current law to the above circuits along with Ohm’s law and the rules for combining resistors gives the numbers shown below. The determining of the voltages and currents associated with a particular circuit along with the power allows you to completely describe the electrical state of a direct current circuit.
Electric Field
Electric field is defined as the electric force per unit charge. The direction of the field is taken to be the direction of the force it would exert on a positive test charge. The electric field is radially outward from a positive charge and radially in toward a negative point charge.
Click on any of the examples above for more detail.
Electric Field of Point Charge
The electric field of a point charge can be obtained from Coulomb’s law:
The electric field is radially outward from the point charge in all directions. The circles represent spherical equipotential surfaces. 
The electric field from any number of point charges can be obtained from a vector sum of the individual fields. A positive number is taken to be an outward field; the field of a negative charge is toward it.
This electric field expression can also be obtained by applying Gauss’ law.
Electric and Magnetic Constants
In the equations describing electric and magnetic fields and their propagation, three constants are normally used. One is the speed of light c, and the other two are the electric permittivity of free space ε_{0} and the magnetic permeability of free space, μ_{0}. The magnetic permeability of free space is taken to have the exact value
See also relative permeability 
This contains the force unit N for Newton and the unit A is the Ampere, the unit of electric current.
With the magnetic permeability established, the electric permittivity takes the value given by the relationship
where the speed of light c is given by
This gives a value of free space permittivity
which in practice is often used in the form
These expressions contain the units F for Farad, the unit of capacitance, and C for Coulomb, the unit of electric charge.
In the presence of polarizable or magnetic media, the effective constants will have different values. In the case of a polarizable medium, called a dielectric, the comparison is stated as a relative permittivity or a dielectric constant. In the case of magnetic media, the relative permeability may be stated.
Physical Connections to Electric Permittivity and Magnetic Permeability
Expressions for the electric and magnetic fields in free space contain the electric permittivity ε_{0} and magnetic permeability μ_{0} of free space. As indicated in the section on electric and magnetic constants, these two quantities are not independent but are related to “c”, the speed of light and other electromagnetic waves.
The electric permittivity is connected to the energy stored in an electric field. It is involved in the expression for capacitance because it affects the amount of charge which must be placed on a capacitor to achieve a certain net electric field. In the presence of a polarizable medium, it takes more charge to achieve a given net electric field and the effect of the medium is often stated in terms of a relative permittivity. 
The magnetic permeability is connected to the energy stored in a magnetic field. It is involved in the expression for inductance because in the presence of a magnetizable medium, a larger amount of energy will be stored in the magnetic field for a given current through the coil. The effect of the medium is often stated in terms of a relative permeability. 
Current Law and Flowrate
For any circuit, fluid or electric, which has multiple branches and parallel elements, the flowrate through any crosssection must be the same. This is sometimes called the principle of continuity.
Voltage Law and Pressure
Ohm’s LawPoiseuille’s Law
Ohm’s law for electric current flow and Poiseuille’s law for the smooth flow of fluids are of the same form.
Will the bird on the high voltage wire be shocked?
Electric current flow is proportional to voltage difference according to Ohm’s law, and both the bird’s feet are at the same voltage. Since current flow is necessary for electric shock, the bird is quite safe unless it simultaneously touches another wire with a different voltage.
Want a scary job? Maintenance on high voltage transmission lines is sometimes done with the voltage “live” by working from a platform on a helicopter, sitting on a metal platform! The helicopter must make sure it doesn’t touch neighboring wires which are at a different voltage. 
Basic DC Circuit Relationships
DC circuits can be completely analyzed with these four relationships.  Ohm’s law  I = V/R 
Power relationship  P = VI  
Voltage Law  The net voltage change is equal to zero around any closed loop. (This is an application of the principle of conservation of energy.)  
Current Law  The electric current in = electric current out of any junction. (Conservation of charge) 
Norton’s Theorem
Any collection of batteries and resistances with two terminals is electrically equivalent to an ideal current source i in parallel with a single resistor r. The value of r is the same as that in the Thevenin equivalent and the current i can be found by dividing the open circuit voltage by r.
Norton Current
The value i for the current used in Norton’s Theorem is found by determining the open circuit voltage at the terminals AB and dividing it by the Norton resistance r.
Norton Example
Replacing a network by its Norton equivalent can simplify the analysis of a complex circuit. In this example, the Norton current is obtained from the open circuit voltage (the Thevenin voltage) divided by the resistance r. This resistance is the same as the Thevenin resistance, the resistance looking back from AB with V_{1} replaced by a short circuit.
//
For R_{1} = Ω, R_{2} = Ω, R_{3} = Ω, 
and voltage V_{1} = V 
the open circuit voltage is  = V 
since R_{1} and R_{3} form a simple voltage divider. 
The Norton resistance is  = Ω. 
and the resulting Norton current is  = A 
Superposition: Two Loop Problem
To apply the superposition theorem to calculate the current through resistor R_{1} in the two loop circuit shown, the individual current supplied by each battery is calculated with the other battery replaced by a short circuit.
//
For R_{1} =Ω, R_{2} = Ω, R_{3} = Ω, 
and voltages V_{1} = V and V_{2} = V, 
the calculated currents are
= A,  = A 
with a resultant current in R_{1} of  = A. 
Note: To avoid dealing with so many short circuits, any resistor with value zero will default to 1 when a voltage is changed. It can be changed back to a zero value if you wish to explore the effects of short circuits. Ohms and amperes are the default units, but if you put in resistor values in kilohms, then the currents will be milliamperes.
Superposition Theorem
The total current in any part of a linear circuit equals the algebraic sum of the currents produced by each source separately.
To evaluate the separate currents to be combined, replace all other voltage sources by short circuits and all other current sources by open circuits.
Thevenin’s Theorem
Any combination of batteries and resistances with two terminals can be replaced by a single voltage source e and a single series resistor r. The value of e is the open circuit voltage at the terminals, and the value of r is e divided by the current with the terminals short circuited.
Thevenin Voltage
The Thevenin voltage e used in Thevenin’s Theorem is an ideal voltage source equal to the open circuit voltage at the terminals. In the example below, the resistance R_{2} does not affect this voltage and the resistances R_{1} and R_{3} form a voltage divider, giving
Thevenin/Norton Resistance
The Thevenin resistance r used in Thevenin’s Theorem is the resistance measured at terminals AB with all voltage sources replaced by short circuits and all current sources replaced by open circuits. It can also be calculated by dividing the open circuit voltage by the short circuit current at AB, but the previous method is usually preferable and gives
The same resistance is used in the Norton equivalent.
Thevenin Example
Replacing a network by its Thevenin equivalent can simplify the analysis of a complex circuit. In this example, the Thevenin voltage is just the output of the voltage divider formed by R_{1} and R_{3}. The Thevenin resistance is the resistance looking back from AB with V_{1} replaced by a short circuit.
//
For R_{1} =Ω, R_{2}  = Ω, R_{3} =Ω, 
and voltage V_{1} = V 
the Thevenin voltage is  V 
since R_{1} and R_{3} form a simple voltage divider. 
The Thevenin resistance is  = Ω. 
Two Loop Currents
Application of the voltage law and the current law. This approach requires two simultaneous equations formed by applying the voltage law to both loops. Taking the loops in the directions shown means that the sum of the currents flows through the middle resistor. Putting polarities on each component helps to keep the signs straight. The two loop equations are:
Subtracting (2) from (1) and solving for the currents gives: 
//
For R_{1} =Ω, R_{2} =  Ω, R_{3} =  Ω, 
and voltages V_{1} = V and V_{2} = V, 
the calculated currents are
= A,  = A. 
Note: To avoid dealing with so many short circuits, any resistor with value zero will default to 1 when a voltage is changed. It can be changed back to a zero value if you wish to explore the effects of short circuits. Ohms and amperes are the default units, but if you put in resistor values in kilohms, then the currents will be milliamperes.
Strategy for a Two Loop Circuit
If you know one current in a twoloop circuit, say by measuring it with an ammeter, then the circuit can be simplified using the voltage law and the current law. This approach avoids the two simultaneous equations required for the general case.

1. Note that the right hand branch must be analyzed first since it is the only one with full information. Find the voltage V at the top of the circuit.
Note by the voltage law that V is the voltage for all three parallel branches. 2. The left branch mst be analyzed next since the center branch still has two unknowns. Find the branch current from 3. From the current law applied to the junction at top center Then find the unknown voltage from 
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