Physics for Science & Engineering II
Physics for Science & Engineering II
By Yildirim Aktas, Department of Physics & Optical Science
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  • Introduction
  • Syllabus
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    • Chapter 01: Electric Charge
      • 1.1 Fundamental Interactions
      • 1.2 Electrical Interactions
      • 1.3 Electrical Interactions 2
      • 1.4 Properties of Charge
      • 1.5 Conductors and Insulators
      • 1.6 Charging by Induction
      • 1.7 Coulomb Law
        • Example 1: Equilibrium Charge
        • Example 2: Three Point Charges
        • Example 3: Charge Pendulums
    • Chapter 02: Electric Field
      • 2.1 Electric Field
      • 2.2 Electric Field of a Point Charge
      • 2.3 Electric Field of an Electric Dipole
      • 2.4 Electric Field of Charge Distributions
        • Example 1: Electric field of a charged rod along its Axis
        • Example 2: Electric field of a charged ring along its axis
        • Example 3: Electric field of a charged disc along its axis
        • Example 4: Electric field of a charged infinitely long rod.
        • Example 5: Electric field of a finite length rod along its bisector.
      • 2.5 Dipole in an External Electric Field
    • Chapter 03: Gauss’ s Law
      • 3.1 Gauss’s Law
        • Example 1: Electric field of a point charge
        • Example 2: Electric field of a uniformly charged spherical shell
        • Example 3: Electric field of a uniformly charged soild sphere
        • Example 4: Electric field of an infinite, uniformly charged straight rod
        • Example 5: Electric Field of an infinite sheet of charge
        • Example 6: Electric field of a non-uniform charge distribution
      • 3.2 Conducting Charge Distributions
        • Example 1: Electric field of a concentric solid spherical and conducting spherical shell charge distribution
        • Example 2: Electric field of an infinite conducting sheet charge
      • 3.3 Superposition of Electric Fields
        • Example: Infinite sheet charge with a small circular hole.
    • Chapter 04: Electric Potential
      • 4.1 Potential
      • 4.2 Equipotential Surfaces
        • Example 1: Potential of a point charge
        • Example 2: Potential of an electric dipole
        • Example 3: Potential of a ring charge distribution
        • Example 4: Potential of a disc charge distribution
      • 4.3 Calculating potential from electric field
      • 4.4 Calculating electric field from potential
        • Example 1: Calculating electric field of a disc charge from its potential
        • Example 2: Calculating electric field of a ring charge from its potential
      • 4.5 Potential Energy of System of Point Charges
      • 4.6 Insulated Conductor
    • Chapter 05: Capacitance
      • 5.01 Introduction
      • 5.02 Capacitance
      • 5.03 Procedure for calculating capacitance
      • 5.04 Parallel Plate Capacitor
      • 5.05 Cylindrical Capacitor
      • 5.06 Spherical Capacitor
      • 5.07-08 Connections of Capacitors
        • 5.07 Parallel Connection of Capacitors
        • 5.08 Series Connection of Capacitors
          • Demonstration: Energy Stored in a Capacitor
          • Example: Connections of Capacitors
      • 5.09 Energy Stored in Capacitors
      • 5.10 Energy Density
      • 5.11 Example
    • Chapter 06: Electric Current and Resistance
      • 6.01 Current
      • 6.02 Current Density
        • Example: Current Density
      • 6.03 Drift Speed
        • Example: Drift Speed
      • 6.04 Resistance and Resistivity
      • 6.05 Ohm’s Law
      • 6.06 Calculating Resistance from Resistivity
      • 6.07 Example
      • 6.08 Temperature Dependence of Resistivity
      • 6.09 Electromotive Force, emf
      • 6.10 Power Supplied, Power Dissipated
      • 6.11 Connection of Resistances: Series and Parallel
        • Example: Connection of Resistances: Series and Parallel
      • 6.12 Kirchoff’s Rules
        • Example: Kirchoff’s Rules
      • 6.13 Potential difference between two points in a circuit
      • 6.14 RC-Circuits
        • Example: 6.14 RC-Circuits
    • Chapter 07: Magnetism
      • 7.1 Magnetism
      • 7.2 Magnetic Field: Biot-Savart Law
        • Example: Magnetic field of a current loop
        • Example: Magnetic field of an infinitine, straight current carrying wire
        • Example: Semicircular wires
      • 7.3 Ampere’s Law
        • Example: Infinite, straight current carrying wire
        • Example: Magnetic field of a coaxial cable
        • Example: Magnetic field of a perfect solenoid
        • Example: Magnetic field of a toroid
        • Example: Magnetic field profile of a cylindrical wire
        • Example: Variable current density
    • Chapter 08: Magnetic Force
      • 8.1 Magnetic Force
      • 8.2 Motion of a charged particle in an external magnetic field
      • 8.3 Current carrying wire in an external magnetic field
      • 8.4 Torque on a current loop
      • 8.5 Magnetic Domain and Electromagnet
      • 8.6 Magnetic Dipole Energy
      • 8.7 Current Carrying Parallel Wires
        • Example 1: Parallel Wires
        • Example 2: Parallel Wires
    • Chapter 09: Induction
      • 9.1 Magnetic Flux, Fraday’s Law and Lenz Law
        • Example: Changing Magnetic Flux
        • Example: Generator
        • Example: Motional emf
        • Example: Terminal Velocity
        • Simulation: Faraday’s Law
      • 9.2 Induced Electric Fields
      • Inductance
        • 9.3 Inductance
        • 9.4 Procedure to Calculate Inductance
        • 9.5 Inductance of a Solenoid
        • 9.6 Inductance of a Toroid
        • 9.7 Self Induction
        • 9.8 RL-Circuits
        • 9.9 Energy Stored in Magnetic Field and Energy Density
      • Maxwell’s Equations
        • 9.10 Maxwell’s Equations, Integral Form
        • 9.11 Displacement Current
        • 9.12 Maxwell’s Equations, Differential Form
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Online Lectures » Chapter 06: Electric Current and Resistance » 6.06 Calculating Resistance from Resistivity

6.06 Calculating Resistance from Resistivity

6.6 Calculating Resistance from Resistivity from Office of Academic Technologies on Vimeo.

6.06 Calculating Resistance from Resistivity

Since both of these quantities, resistance and resistivity, are in a way linked to the number of collisions that the charge carriers are making as they drift from high-potential region to low-potential region, then we can expect a relationship between these two quantities. To be able to show this relationship, let’s consider a piece of wire with a length of, say, l. And let’s connect the ends of this wire to a power supply, which generates V volts of potential difference between its terminals. Therefore, as soon as we turn the switch on we’re going to generate V volts of potential difference between these two ends of this wire. And, again, at the moment that we turn the switch on, we will set up an electric field pointing from positive end of this wire towards the negative end.

Let’s say that the cross-sectional area of the wire is A. So, A represents the cross-sectional area. The potential difference between the ends of this wire will be equal to, as you recall, integral of E dot dl, integrated along the length of this wire. Well, if you do this, since the electric field magnitude is constant, and by choosing a path from one end to the other end, dl represents the incremental displacement vector along this path. So the angle between these two vectors, vector-field vector times the incremental displacement field vector dl times the cosine of the angle between these two vectors — in this case it’s going to be 0 degree — will give us the expanded form of E dot dl.

Again, since E is constant, we can take it outside of the integral, and cosine of 0 is just 1, this quantity is going to be equal to E times integral of dl along the length of this wire, and its length — let’s say we’ll put our origin at one end — then it’s going to go from 0 to l, so the integral of dl is addition of these incremental distances, dl‘s to one another along the length of the wire, will give us whatever the length of that wire is. So the potential difference is going to be equal to the electric field along the wire times its length, l.

From there we can solve for the electric field, which will be equal to V divided by the potential difference between the ends of the wire divided by its length. On the other hand, we know that the current density, J, is equal to current flowing through the wire, divided by the cross-sectional area of this wire.

Now, by recalling the definition of resistivity, which was the ratio of the electric field to the current density, we can express these quantities as V over l for the electric field divided by i over A for the current density. Going one further step, this is going to be equal to V over i times A over l. Well, V over i , by definition, that is the potential difference between the ends of this wire, V, divided by the amount of current flowing through this wire, and that is i, which is going to originate from the positive terminal and enter into the negative terminal.

This quantity is therefore nothing but, by definition, just the resistance, R, of this wire. So, we end up with R times A over l. If we solve this expression for the resistance, then that becomes equal to resistivity ρ times l over a. That’s the relationship between resistance and resistivity for a wire with length l and cross-sectional area A. From here we can easily see that the resistance is directly proportional to the length of the wire. In other words, the longer the wire means the greater the resistance is, which will eventually cause resistive losses that we will study in a moment.

So in order to avoid resistive losses — in other words, in order to avoid the waste of electrical potential energy eventually in the form of heat due to this resistance — as we design our circuits we’d like to keep the length of the wires as short as possible. On the other hand, we see that the resistance is inversely proportional to the cross-sectional area of the wire. So that tells us that the thicker the wires will cause less amount of resistive losses. But of course when we do the physical calculations that doesn’t mean that we just take our wires very, very thick so that we will decrease the resistive losses because such a process will even cause more in comparing to the amount of electrical potential energy wasted in the form of heat.

So the major part that we have to be careful is the length of the wires, and we try to keep them as short as possible in order to decrease the resistive losses in the electric circuits. And of course, as we expect, the resistance is directly proportional to the resistivity because both of these quantities are, in a way, measure of the number of collisions that the charge carriers are making as they drift from high-potential regions to low-potential regions.

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