Chapter 23

Nature is symmetric, prompting scientists to investigate the relationship between magnetic fields and electric currents.
Key Historical Figures: Michael Faraday and Joseph Henry
- Faraday (1791–1862) and Henry (1797–1878) independently demonstrated that magnetic fields could produce electric currents (1831).
Experimental Apparatus Used by Faraday:
- A coil and a galvanometer were utilized to illustrate that the movement of a magnet relative to a coil generates electromotive force (emf).

Faraday's Law of Induction:
- The induced emf (ε) is proportional to the negative rate of change of magnetic flux (Φ_m) through a circuit:
  $ext{Induced emf: } ext{ε} = - rac{dΦ_m}{dt}$
Magnetic Flux (Φ):
- Defined as the total number of magnetic field lines passing through a loop, given by:
  $ext{Φ} = B imes A imes ext{cos}(θ)$
- Units of magnetic flux are the weber (Wb):
  $1 ext{ Wb} = 1 ext{ T m}^2$
Lenz's Law:
- States that the direction of the induced emf and induced current will oppose the change in magnetic flux that produced it.
- Always leads to a direction of current that restores the original magnetic field configuration, maintaining conservation of energy.

Example 1: A Square Coil in a Changing Magnetic Field
- Square coil with side length $l = 0.25 ext{ m}$ and $N = 200$ turns of wire.
- Resistance of the coil $R = 5.0 ext{ Ω}$ .
- Uniform magnetic field decreasing at a rate $rac{ΔB}{Δt} = -0.040 ext{ T/s}$ .
- (a) Calculate induced emf.
- (b) Calculate current using Ohm's Law: $V = IR$ .
Example 2: Solenoid with Coil
- Solenoid with $n = 10$ turns/cm, area $= 5.0 ext{ cm}^2$ , and current $I = 0.25 ext{ A}$ .
- Average emf induced in a nearby coil with 5 turns when solenoid current decreases to zero in 0.050 s.
Example 3: Copper Wire Loop
- A square loop with sides 6.0 cm made of copper wire with radius 1.0 mm. Magnetic field changing at a rate of $5.0 ext{ mT/s}$ .
- Find the current using previous laws and given $<br>ho = 1.68 imes 10^{-8} ext{ Ω m}$ for resistivity of copper.
Example 4: Earth's Magnetic Field
- Find the magnetic flux through a square loop of area $20.0 ext{ cm}^2$ in Earth's magnetic field of $5.00 imes 10^{-5} ext{ T}$ at different angles: perpendicular, 30°, and 90° to the loop plane.
Example 5: Reversing Magnetic Field
- A circular coil with resistance 5.0 Ω enclosing an area of 100 cm² initially in a uniform field of $1.1 ext{ T}$ upwards, which reverses direction in 0.10 s.
- Explore average current induced during the reversal.

When a conductor moves through a magnetic field, an emf is induced.
- Motional EMF Formula: $ext{emf} = B imes ext{ℓ} imes v$ where:
- $B$ = magnetic field strength,
- $ext{ℓ}$ = length of the conductor,
- $v$ = velocity of movement.
Example Calculation:
- For a 1 m rod moving at 3.0 m/s in a magnetic field of $5.0 imes 10^{-5} ext{ T}$ :
  $ext{emf} = (5.0 imes 10^{-5} ext{ T}) (1.0 ext{ m})(3.0 ext{ m/s}) = 150 ext{ μV}$ .
Example 6: Airplane EMF
- An airplane traveling at 1000 km/h in a region with a vertical magnetic field: What is the potential difference induced between wing tips 70 m apart?
- Calculations show a very small voltage induced ( $ext{approx. } 1 ext{ V}$ ).
Example 7: Wire in a Magnetic Field
- A 2.00 m length of wire moves at 15.0 m/s through a vertical magnetic field of 40.0 µT downward. Determine induced emf and positive end.
Example 8: Astronaut Conducting Voltage
- An astronaut's tether induces a measured voltage of 0.45 V in a wire as they orbit Earth. Explore the change in measured voltage with movement.

Electric generators convert mechanical energy to electrical energy through the rotation of coils within magnetic fields.
- Basic Principles: Same as motors, but operate in reverse.
Induced emf in Generators:
- Generated emf varies sinusoidally over time as coils rotate, maximum emf occurs when coiling orientation is best aligned with magnetic field (Sin(θ) = 1).
Output Characteristics:
- The system includes rings and brushes, with the produced emf often in alternating current (AC) format.
Example 9: A coil rotating at 60 rev/s in a magnetic field can induce a certain voltage, solved via the equation including turns and area.

Transformers adapt electrical voltage levels for various applications.
Transformer Operation:
- Step-Up Transformers: Increase voltage, while Step-Down Transformers: decrease voltage.
Power Conservation: For an ideal transformer, power input equals power output:
$P_{in} = P_{out}$
Transformer Equations:
- Voltage and turns ratio:
  $rac{V_p}{V_s} = rac{N_p}{N_s}$ ,
- Current relevance:
  $I_p V_p = I_s V_s$ .
Example of Step-Up Transformer: An x-ray unit transforms 120 V input to a high voltage output.
Example of Step-Down Transformer: Voltage from a battery charger computes required loops in the secondary coil and relationship to the needed input current.