Ohm's Law, Resistance, and Resistivity Factors - Tenth Grade Science
The relationship between the potential difference applied across an electrical conductor and the resulting current that flows through it forms one of the central pillars of circuit theory. Formulated experimentally by the German physicist Georg Simon Ohm in 1827, Ohm's law defines the property of electrical resistance and explains how geometric dimensions, material composition, and environmental temperature dictate the conductivity of materials.
Ohm's Law
Ohm's law describes the mathematical proportionality between current and voltage across an ideal metallic conductor maintained under constant physical conditions.
EXPERIMENTAL VERIFICATION CIRCUIT
Rheostat (Variable Resistor)
[====\====]
+-----------------+------------------+
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--- DC Source |
----- (Battery) |
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+-------------(A) ammeter -----------+
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+---+---+
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[Conductor R]
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+--(V)--+
voltmeter
Statement of the Law
Ohm's law states that the electric current ($I$) flowing through a metallic conductor is directly proportional to the potential difference ($V$) applied across its terminal ends, provided its temperature and other physical parameters remain strictly constant.
Mathematically:
$$V \propto I \implies \frac{V}{I} = \text{Constant} = R$$
Rearranging this expression gives the primary algebraic formulations of Ohm's law:
$$V = I R \quad \text{or} \quad I = \frac{V}{R} \quad \text{or} \quad R = \frac{V}{I}$$
The Concept of Electrical Resistance
The constant of proportionality $R$ is known as the electrical resistance of the conductor. Resistance represents the opposition that a material body presents to the flow of electric charge.
At the microscopic level, free conduction electrons accelerated by an applied electric field continually collide with vibrating metal cations in the crystal lattice. These collisions impede the forward drift velocity of the electrons, dissipating electrical potential energy into microscopic thermal vibrations (heat). A material that allows charges to move with minimal collisions exhibits low resistance, while a material that induces frequent collisions exhibits high resistance.
V-I CHARACTERISTIC (OHMIC CONDUCTOR)
Voltage (V)
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| / Slope = R = Delta V / Delta I
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0-----+-----------------> Current (I)
Straight linear line passing through origin:
Confirms Constant Resistance (Ohmic Behavior)
The SI Unit of Resistance: The Ohm
The SI unit of electrical resistance is the ohm, symbolized by the Greek capital letter omega ($\Omega$).
According to Ohm's law ($R = V / I$), one ohm is defined as the resistance of a conductor through which a steady current of one ampere flows when an electric potential difference of one volt is maintained across its ends:
$$1\ \Omega = \frac{1 \text{ V}}{1 \text{ A}} = 1 \text{ V/A}$$
Ohmic and Non-Ohmic Conductors
Not all electrical materials and components obey Ohm's law across their operating range.
OHMIC VS NON-OHMIC V-I CHARACTERISTICS
OHMIC (Copper Wire) NON-OHMIC (Semiconductor Diode)
V V
^ / ^ /
| / | /
| / | /
| / | ______/ (Threshold)
+-----------> I +-----------> I
Ohmic Conductors
Conductors that strictly satisfy the linear proportionality $V \propto I$ are termed ohmic conductors. Common examples include pure metallic wires (such as copper, silver, and gold) and metallic alloy resistors held at constant temperature. Their current-voltage graph yields a straight line passing through the coordinate origin.
Non-Ohmic Conductors
Conductors in which current does not scale linearly with applied voltage are termed non-ohmic conductors. Examples include semiconductor diodes, transistors, electrolyte solutions, vacuum tubes, and incandescent light bulb filaments. In an incandescent bulb, the tungsten filament heats up to over 2500 degrees Celsius during operation. As the temperature rises, lattice vibrations increase, causing resistance to climb. Consequently, the $V$-$I$ curve bends upward, demonstrating non-linear, non-ohmic behavior.
Factors Influencing the Resistance of a Conductor
Systematic laboratory investigations demonstrate that the electrical resistance ($R$) of an isotropic cylindrical conductor depends on four distinct physical variables.
FACTORS AFFECTING WIRE RESISTANCE
<--------------- Length (L) --------------->
+------------------------------------------+
Cross- | |
Section (A) | Conductor Body |
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+------------------------------------------+
1. Resistance is proportional to Length: R ~ L
2. Resistance is inverse to Area: R ~ 1/A
3. Resistance depends on Material Nature: rho
4. Resistance increases with Temperature: T
1. Dependence on Length of Conductor ($L$)
The resistance of a uniform conductor is directly proportional to its length ($R \propto L$). If the length of a wire is doubled, conduction electrons must travel twice the distance, encountering twice as many lattice collisions. Consequently, its resistance doubles.
2. Dependence on Cross-Sectional Area ($A$)
The resistance of a conductor is inversely proportional to its cross-sectional area ($R \propto 1/A$). A thicker wire provides a wider cross-section containing a greater count of parallel conduits for free electrons, reducing congestion and collisions. Doubling the cross-sectional area halves the total resistance.
3. Dependence on Material Composition: Resistivity ($\rho$)
Combining the geometric proportionalities yields:
$$R \propto \frac{L}{A} \implies R = \rho \frac{L}{A}$$
The constant of proportionality $\rho$ (the Greek letter rho) is an intrinsic property of the material known as electrical resistivity, or specific resistance.
The SI Unit and Nature of Resistivity
Rearranging the resistance formula to solve for resistivity gives:
$$\rho = R \frac{A}{L}$$
Substituting standard SI units into the expression:
$$[\rho] = \Omega \times \frac{\text{m}^2}{\text{m}} = \Omega\cdot\text{m} \quad (\text{ohm-meter})$$
The SI unit of resistivity is the ohm-meter ($\Omega\cdot\text{m}$).
Resistivity is an intrinsic material property that is completely independent of the conductor's physical dimensions (length or cross-sectional area). A copper wire 10 meters long has identical resistivity to a copper wire 1 millimeter long, although their total resistances differ substantially.
REPRESENTATIVE RESISTIVITY VALUES
Classification Material Resistivity (rho at 20 deg C)
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Conductors Silver 1.60 x 10^-8 Ohm*m
Copper 1.68 x 10^-8 Ohm*m
Aluminum 2.65 x 10^-8 Ohm*m
Tungsten 5.60 x 10^-8 Ohm*m
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Alloys Manganin 4.40 x 10^-7 Ohm*m
Constantan 4.90 x 10^-7 Ohm*m
Nichrome 1.10 x 10^-6 Ohm*m
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Insulators Glass 10^10 to 10^14 Ohm*m
Hard Rubber 10^13 to 10^16 Ohm*m
Fused Quartz ~10^17 Ohm*m
4. Dependence on Temperature
For pure metals, increasing the temperature increases the kinetic energy and vibrational amplitude of the lattice cations. The drifting electrons collide more frequently with the agitated atoms, which increases electrical resistance and resistivity. For metals, the temperature coefficient of resistance is positive.
For standard alloys like Nichrome, Constantan, and Manganin, the resistivity is noticeably higher than that of their constituent pure metals, while their temperature coefficient of resistance is negligible. Their resistance remains virtually unchanged over wide temperature variations. This property makes alloys ideal for manufacturing standard laboratory resistors, rheostats, and electrical heating elements.
Conductance and Conductivity
To express how easily electric current flows through materials, reciprocal properties are used.
Electrical Conductance ($G$)
Conductance is defined as the reciprocal of electrical resistance:
$$G = \frac{1}{R}$$
The SI unit of conductance is the siemens ($\text{S}$), also historically designated as the mho ($\Omega^{-1}$).
Electrical Conductivity ($\sigma$)
Conductivity is defined as the reciprocal of electrical resistivity:
$$\sigma = \frac{1}{\rho}$$
The SI unit of electrical conductivity is siemens per meter ($\text{S/m}$) or $\Omega^{-1}\cdot\text{m}^{-1}$. Materials with high conductivity (like silver and copper) serve as transmission wiring, while materials with near-zero conductivity serve as electrical insulation.
Chapter Exercises and Solutions
Exercise 1: A uniform cylindrical metallic wire having a resistance of $16\ \Omega$ is drawn through a wire die such that its length is stretched to exactly twice its original value while its total volume remains strictly conserved. Calculate the new electrical resistance of the stretched wire.
Solution 1: Let the initial length, cross-sectional area, and resistance of the wire be denoted by $L_1$, $A_1$, and $R_1 = \rho L_1 / A_1 = 16\ \Omega$. When the wire is stretched to double its original length, the new length is $L_2 = 2 L_1$. Because the volume of the metallic material is conserved during drawing ($V = A_1 L_1 = A_2 L_2$), doubling the length forces the cross-sectional area to decrease by half: $A_2 = A_1 L_1 / L_2 = A_1 / 2$. Expressing the new resistance $R_2$ using these dimensions gives $R_2 = \rho L_2 / A_2 = \rho (2 L_1) / (A_1 / 2) = 4 (\rho L_1 / A_1) = 4 R_1$. Substituting the original resistance value yields $R_2 = 4 \times 16\ \Omega = 64\ \Omega$. Stretching the wire to twice its length increases its resistance by a factor of four, yielding $64\ \Omega$.
Exercise 2: A copper wire of length 2.0 meters and circular cross-sectional radius 0.50 millimeters exhibits an electrical resistance of $0.043\ \Omega$. Compute the electrical resistivity of copper based on these dimensions.
Solution 2: First, convert the given dimensions into standard SI units. The length is $L = 2.0 \text{ m}$. The radius is $r = 0.50 \text{ mm} = 0.50 \times 10^{-3} \text{ m}$. The cross-sectional area $A$ of the circular wire is given by $A = \pi r^2 = \pi \times (0.50 \times 10^{-3} \text{ m})^2 = \pi \times 2.5 \times 10^{-7} \text{ m}^2 \approx 7.854 \times 10^{-7} \text{ m}^2$. The formula for resistivity is $\rho = R A / L$. Substituting the given resistance and geometric values yields $\rho = (0.043\ \Omega \times 7.854 \times 10^{-7} \text{ m}^2) / 2.0 \text{ m} \approx 1.69 \times 10^{-8}\ \Omega\cdot\text{m}$. The measured resistivity of copper is approximately $1.69 \times 10^{-8}\ \Omega\cdot\text{m}$, which aligns with accepted experimental standards.
Exercise 3: Explain why electrical heating appliances such as toasters, room heaters, and electric irons utilize elements constructed from Nichrome alloy rather than pure copper or silver wire, citing two distinct physical properties.
Solution 3: Nichrome (an alloy composed of nickel, chromium, manganese, and iron) is preferred over pure copper or silver for heating elements due to two main physical factors. First, Nichrome has an electrical resistivity roughly 60 times higher than that of copper. According to Joule's law of heating ($H = I^2 R t$), high resistance allows a compact length of Nichrome wire to generate substantial thermal energy at moderate currents, whereas a copper wire would require extreme lengths to produce equivalent heat. Second, Nichrome has a high melting point and does not easily oxidize or burn at red-hot operating temperatures (around $900^\circ\text{C}$ to $1100^\circ\text{C}$). Pure copper or silver wires would oxidize rapidly in atmospheric air, become brittle, and melt at high temperatures.