Chapter 14: Electromagnetism

• Magnetic Field (B)

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  • Region around a magnet or current-carrying conductor where magnetic force is exerted.
  • Magnetic field strength (Magnetic Induction / Magnetic Flux Density).
  • Unit: Tesla (T) or Weber per square meter ($Wb/m^2$).
  • $1~T = 10^4~Gauss$.
  • Dimensions: $[MT^{-2}A^{-1}]$.
  • Vector quantity.

  • Magnetic Field Lines

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    • Originate from North pole, terminate on South pole (outside magnet).
    • Form closed loops (inside magnet, South to North).
    • Tangent at any point gives direction of $\vec{B}$.
    • Density of lines indicates strength of $\vec{B}$.
    • Never cross each other.

• Force on a Current-Carrying Conductor in a Magnetic Field

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  • Force experienced by a conductor carrying current in a magnetic field.
  • $\vec{F} = I (\vec{L} \times \vec{B})$.
  • Magnitude: $F = I L B \sin\theta$.
  • Direction by Right-Hand Rule (for current) or Fleming's Left-Hand Rule.
  • Maximum force when $\theta = 90^\circ$ ($F = ILB$).
  • Zero force when $\theta = 0^\circ$ or $180^\circ$.

• Force on a Charge Moving in a Magnetic Field (Lorentz Force)

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  • Force experienced by a moving charge in a magnetic field.
  • $\vec{F} = q (\vec{v} \times \vec{B})$.
  • Magnitude: $F = q v B \sin\theta$.
  • Direction by Right-Hand Rule (for positive charge) or Fleming's Left-Hand Rule.
  • Maximum force when $\theta = 90^\circ$ ($F = qvB$).
  • Zero force when $\theta = 0^\circ$ or $180^\circ$.
  • Magnetic force does no work on the charge (force perpendicular to velocity).
  • If charge moves perpendicular to field, it follows a circular path.
  • Radius of circular path: $r = \frac{mv}{qB}$.
  • Frequency of revolution: $f = \frac{qB}{2\pi m}$ (cyclotron frequency).

• Magnetic Flux ($\Phi_B$)

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  • Measure of the number of magnetic field lines passing through a given area.
  • $\Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta$.
  • Unit: Weber (Wb).
  • Scalar quantity.

• Ampere's Law

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  • Line integral of magnetic field around any closed loop is proportional to the total current enclosed by the loop.
  • $\oint \vec{B} \cdot d\vec{l} = \mu_o I_{enclosed}$.
  • $\mu_o$: permeability of free space ($4\pi \times 10^{-7}~T m A^{-1}$).
  • Used to calculate magnetic field for symmetric current distributions.

  • Applications of Ampere's Law

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    • Magnetic Field due to a Long Straight Current-Carrying Wire: $B = \frac{\mu_o I}{2\pi r}$.
    • Magnetic Field inside a Solenoid: $B = \mu_o n I$. ($n$: number of turns per unit length).
    • Magnetic Field inside a Toroid: $B = \frac{\mu_o N I}{2\pi r}$. ($N$: total turns).

• Magnetic Force between Two Parallel Current-Carrying Wires

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  • Force per unit length: $\frac{F}{L} = \frac{\mu_o I_1 I_2}{2\pi d}$.
  • $d$: distance between wires.
  • Attraction if currents are in same direction.
  • Repulsion if currents are in opposite directions.
  • Used to define the Ampere.

• Torque on a Current Loop in a Magnetic Field

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  • A current loop in a uniform magnetic field experiences a torque.
  • $\vec{\tau} = \vec{\mu} \times \vec{B}$.
  • Magnitude: $\tau = NIAB \sin\alpha$.
  • $N$: number of turns.
  • $I$: current.
  • $A$: area of loop.
  • $\alpha$: angle between magnetic dipole moment ($\vec{\mu}$) and magnetic field ($\vec{B}$).
  • Magnetic dipole moment: $\vec{\mu} = NI\vec{A}$.
  • Maximum torque when $\alpha = 90^\circ$ ($\tau_{max} = NIAB$).
  • Zero torque when $\alpha = 0^\circ$ or $180^\circ$.
  • Principle behind electric motors.

• Galvanometer

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  • Device used to detect and measure small electric currents.
  • Works on the principle of torque on a current-carrying coil in a magnetic field.
  • Deflection is proportional to current.

  • Conversion to Ammeter

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    • Low resistance shunt ($R_s$) connected in parallel with galvanometer.
    • $R_s = \frac{I_g R_g}{I - I_g}$.
    • $I_g$: full-scale deflection current of galvanometer.
    • $R_g$: resistance of galvanometer.
    • $I$: desired full-scale current of ammeter.
    • Ammeter has very low resistance, connected in series.

  • Conversion to Voltmeter

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    • High resistance multiplier ($R_m$) connected in series with galvanometer.
    • $R_m = \frac{V}{I_g} - R_g$.
    • $V$: desired full-scale voltage of voltmeter.
    • Voltmeter has very high resistance, connected in parallel.

• Cathode Ray Oscilloscope (CRO)

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  • Electronic instrument used to display varying voltage signals as a two-dimensional graph.
  • Components: Electron gun, Deflecting plates (X and Y), Fluorescent screen.
  • Used for:
    • Measuring AC/DC voltage.
    • Measuring frequency.
    • Phase difference measurement.
    • Waveform analysis.

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