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 08: Magnetic Force » 8.7 Current Carrying Parallel Wires » Example 1: Parallel Wires

Example 1: Parallel Wires

from Office of Academic Technologies on Vimeo.

Example 1- Parallel Wires

Let’s do an example related to the parallel current carrying wires. Let’s assume that we have two parallel wires and from the top view, both of them are carrying a current into the plane direction. Let’s say the first wire’s current is ia. The other one is ib. And these two wires are separated from one another by a distance of d. We’d like to find a point at which the net magnetic field is 0. Find a point such that Btotal at that point is equal to 0.

Okay. First, let’s try to determine the direction of the magnetic field in different regions. If we look at points on the left-hand side of wire a, applying right-hand rule, keep pointing to the right-hand thumb into the plane, and circling the right-hand fingers about the thumb, we will see that the magnetic field lines associated with current a will be rotating in clockwise direction. Therefore on the left-hand side, the magnetic field is going to be tangent to the field line passing through the point of interest. So at this point, it is going to be pointing in upward direction along this line.

Similarly, for ib also, applying the right hand rule, we will see that the field lines are going to be circling in clockwise direction. And the one which is passing through at a point which is on the left-hand side of wire a will also be pointing in upward direction. Since the magnetic fields generated by current a and current b at all points on the left hand side of this wire will be pointing in upward direction, they will never cancel one another and we will never find a point that the net magnetic field will be 0.

On the other hand, if you consider the right-hand side region of ib along the line joining these two wires, and in this case we will see that by applying the right-hand rule, we know that the field lines are circling in clockwise direction for both of them, so ib will generate a magnetic field in downward direction, tangent to the field line, and as well as ia. Therefore, on the right-hand side of current b, also the magnetic fields generated by these two currents will add, and we’re going to end up again, with no point that will have total magnetic field is equal to 0.

So we’re left the region between these two wires. If you consider the points over here, let’s say some that are over here, for example, the magnetic field lines due to current a circling in clockwise direction at the location of this point. Therefore, the magnetic field due to current a will be pointing in downward direction, Ba. And for the current b, again, they’re circling in clockwise direction at the location of this point. The magnetic field associated with current b will be pointing in upward direction. Therefore, this region is the region that we’re looking for, because in these regions, magnetic field lines are going to be aligning in opposite directions due to these two current ia and ib.

Well, whenever they’re equal in magnitude, the we will have the point what we’re trying to find. Let’s assume that at this point, the magnitude of these two magnetic field vectors are equal to one another. Let’s denote this distance as x. Therefore, this distance is going to be equal to d minus x relative to current b.

Well, we know that magnitude-wise Ba is equal to μ0 ia over 2π times the distance between the wire and the point of interest, and that is what we called x. And Bb is equal to μ0 times ib over 2π, the distance between the wire and the point of interest. Let’s say that now that point is P, and that is d minus x.

We want these two magnetic field vectors to be equal to one another so that Btotal is going to be equal to vector sum of Ba and Bb. And since they are in opposite directions, whenever they become equal, when we add them Btotal is going to be equal to 0.

Therefore if we equate these two equations, μ0 times ia over 2π x should be equal to μ0 times ib over 2π d minus x. Here we can cancel μ0’s from both sides and as well as 2π‘s. That leaves us an expression such that ia times d minus x becomes equal to ib times x.

We will rearrange this expression for x. If we open the parentheses on the left-hand side, we will have ia times d minus ia times x is equal to ib times x on the right-hand side. And collecting x terms on one side, ia d will be equal to ia plus ib times x. And solving for x, we’re going to end up with x is equal to ia d over ia plus ib.

Therefore, from the wire a, whenever we’re this much of distance away, then the net magnetic field at that point is going to be equal to 0.

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