The electrical resistance of an object is a measure of its opposition to the passage of a steady electric current. An object of uniform cross section will have a resistance proportional to its length and inversely proportional to its cross-sectional area, and proportional to the resistivity of the material.
Discovered by Georg Ohm in 1827 electrical resistance shares some conceptual parallels with the mechanical notion of friction. The SI unit of electrical resistance is the ohm (Ω).
For a wide variety of materials and conditions, the electrical resistance does not depend on the amount of current through or the potential difference (voltage) across the object, meaning that the resistance R is constant for the given temperature and material. Therefore, the resistance of an object can be defined as the ratio of voltage to current, in accordance with Ohm's law:
A solid conductive metal contains mobile, or free, electrons. These electrons are bound to the metal lattice but not to any individual atom. Even with no external electric field applied, these electrons move about randomly due to thermal energy but, on average, there is zero net current within the metal. Given a plane through which the wire passes, the number of electrons moving from one side to the other in any period of time is on average equal to the number passing in the opposite direction. Thus the interior of a metal is filled up with a large number of unattached electrons that travel aimlessly around like a crowd of displaced persons.
When a metal wire is subjected to electric force applied on its opposite ends, these free electrons rush in the direction of the force, thus forming what we call an electric current. For a steady flow, the current I in amperes can be calculated with the following equation:
where Q is the electric charge in coulombs [C] transferred, and t is the time in seconds (s).
More generally, electric current can be represented as the time rate of change of charge, or
Direct Current resistance
The resistance R of a conductor of uniform cross section can be computed as
where, l is the length of the conductor, measured in metres [m], A is the cross-sectional area of the current flow, measured in square metres [m²] ρ is the electrical resistivity (also called specific electrical resistance) of the material. The SI unit of electrical resistivity is the ohm
metre [Ω m]. Resistivity is a measure of the material's ability to oppose electric current. For practical reasons, any connections to a real conductor will almost certainly mean the current density is not totally uniform. However, this formula still provides a good approximation for long thin conductors such as wires.
Electrical resistivity (also known as specific electrical resistance or volume resistivity) is a measure of how strongly a material opposes the flow of electric current. A low resistivity indicates a material that readily allows the movement of electrical charge.
Electrical resistivity ρ is defined by,
where, ρ is the static resistivity (measured in volt-metres per ampere, V m/A); E is the magnitude of the electric field (measured in volts per metre, V/m); J is the magnitude of the current density (measured in amperes per square metre, A/m²). The electrical resistivity ρ can also be given by,
where R is the electrical resistance of a uniform specimen of the material (measured in ohms, Ω); l is the length of the piece of material (measured in metres, m); A is the cross-sectional area of the specimen (measured in square metres, m²).
Finally, electrical resistivity is also defined as the inverse of the conductivity σ (sigma), of the material, or
The reason resistivity has the units of ohm-metres rather than the more intuitive ohm per metre (Ω/m) can perhaps best be seen by transposing the definition to make resistance the subject;
The resistance of a given sample will increase with the length, but decrease with the cross sectional area. Resistance is measured in ohms. Length over Area has units of 1/distance. To end up with ohms, resistivity must be in the units of "ohms × distance" (SI ohm-metre)
The aim of this experiment is to estimate electrical resistance by using two method:
1) less precise called “technical method”
2) more precise Wheatstone bridge.
Technical Method
This method is directly based on the eq.1. To perform this part of the experiment you would need: voltage source, amperometer and voltameter, set of connecting wires and wirewound resistors with an unknown electrical resistance Rx which must be measured. All these elements we may connect into electrical circuit which is a network that has a closed loop, giving a return path for the current. A number of electrical laws apply to all electrical networks. These include: Kirchhoff's circuit laws are two equalities that deal with the conservation of charge and energy in electrical circuits, and were first described in 1845 by Gustav Kirchhoff.
Kirchhoff's current law:
The sum of all currents entering a node is equal to the sum of all currents leaving the node. At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node. Adopting the convention that every current flowing towards the node is positive and that every current flowing away is negative (or the other way around), this principle can be stated as:
n is the total number of branches with currents flowing towards or away from the node.
Kirchhoff's voltage law:
This law is also called Kirchhoff's second law. The directed sum of the electrical potential differences (voltage) around any closed circuit must be zero. Similarly to Kirchhoff's current law, it can be stated as:
Back to the experimental part. Already mentioned elements voltameter, amperometer, power supply and resistor Rx will be connected into the (A) and (B) type of the electrical circuits.
In the “technical method A” voltameter shows voltage on the measured Rx, so V = Vx, where V – is a voltage read from the voltameter and Vx - is voltage on the resistor Rx.
Amperometer shows current (J) which is the sum of the currents flow both through resistor Rx
(Jx) and voltameter (Jv), so according Kirchhoff's current law:
If Rx<< Rv, then according Kirchhoff's voltage law Jx>>Jv, so we can assume that:
Finally we will obtain:
In the “technical method B” voltameter indicates not only voltage on the resistor Rx, (Vx) but also drop of the voltage on internal resistor of the amperometer, VA;
So, according eq.1 and assumption that J = Jx we will obtain:
where is an internal resistance of the amperometer.
When Rx>>RA we can neglect RA and calculate Rx from the eq.5.
Tasks for the technical method A and B:
1. Please connect circuit A and B, respectively.
2. Please ask supervising person to check electrical circuit.
3. Then please set voltage values and read values of the currents. Please measure at least twice for two different values of the voltage.
4. Please calculate Rx from the eq.5.
Important!!! Please take into account if Rx >> RA or Rx << RA
Table for the technical method A and B:
Method
number of the resistor
J V RA Rv Rx
[A] [V] [ ] [ ] [ ]
A
B
Wheatstone bridge method
For this method we have to have: voltage source, set of connecting wires, wirewound resistors with an unknown electrical resistance Rx which then must be measured, galvanometer (G), resistor with adjustable values (Rz), slat with a stretched wire which then will be divided by the slide D into resistors R1 and R2. Please see scheme below.
The Wheatstone bridge is a measuring instrument invented by Samuel Hunter Christie in 1833 and improved and popularized by Sir Charles Wheatstone in 1843. It is used to measure an unknown electrical resistance Rx by balancing two legs of a bridge circuit, one leg of which includes the unknown component.
Rx is the unknown resistance to be measured; R1, R2 and the resistance of Rz are adjustable. If the ratio of the two resistances in the known leg (R2 / R1) is equal to the ratio of the two in the unknown leg (Rx / Rz), then the voltage between the two midpoints (B and D) will be zero and no current will flow through the galvanometer G. Therefore, if R1, R2 and Rz are known to high precision, then Rx can be measured to high precision. Very small changes in Rx disrupt the balance and are readily detected.
When the voltage between two midpoints B and D is equal zero first, Kirchhoff's first rule is used to find the currents in junctions B and D:
Jx – J1 + JG = 0 JZ – J2 – JG = 0
Then, Kirchhoff's second rule is used for finding the voltage in the loops ABD and BCD:
JxRx – J1R1 + JGRG = 0 JZRZ – J2R2 – JGRG = 0
The bridge is balanced and
J
G= 0
, so the second set of equations can be rewritten as:JxRx = J1R1
JZRZ = J2R2
Then, the equations are divided and rearranged, giving:
From the first rule,
I
z= I
x andI
1= I
2. The desired value ofR
x is now known to be given as:At the point of balance, the ratio of
R
1/ R
2= R
x/ R
zHere, resistances R1 and R2 are resistances of pieces of the wire which is stretched on the wooden bar. Using eq. 3 we obtain:
Tasks for the Wheatstone bridge method 1. Please connect circuit.
2. Please ask supervising person to check electrical circuit.
3. From the method A and B please calculate values of the Rx. These values will be then used to preset the Rz resistance for the Wheatstone bridge method.
4. Then please move slide D to balance the Wheatstone bridge. Please note that galvanometer has adjustable protective resistor R. To read correct values of l1 and l2 the potentiometer of the resistor R has to be set at the minimum!
5. Please calculate Rx values using eq. 6.
Number of resistor
l1 l2 RZ RX
[m] [m] [ ] [ ]
Comparison Table:
6. Please calculate percentage deviations according following formula:
Where Rx (WB) is the value of Rx resistance obtain by the Wheatstone bridge method and Rx (A or B method) is the value of Rx resistance obtain either by A or B technical methods.
Scheme in the technical
method
Number of resistor
PD
[ ] [ ]
A
B
A
B
7. Please write a conclusions according result obtained from the comparison both technical and Wheatstone bridge method.
NAME: DATE:
SURNAME: NUMBER of the team: