LC Oscillations - Examples

  • Example 1:
    • A series LCR circuit has an inductance of 0.2 H, a capacitance of 50 μF, and a resistance of 100 Ω.
    • Calculate the resonant frequency and the quality factor of the circuit.
    • Solution:
      • Resonant frequency:
        • $ f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{0.2 \cdot 10^{-3} \cdot 50 \cdot 10^{-6}}} $ Hz
      • Quality factor:
        • $ Q = \frac{1}{R} \sqrt{\frac{L}{C}} = \frac{1}{100} \sqrt{\frac{0.2}{50 \cdot 10^{-6}}} $
  • Example 2:
    • A capacitor of 10 μF and an inductor of 0.1 H are connected in series.
    • Determine the resonant frequency of the resulting circuit.
    • Solution:
      • Resonant frequency:
        • $ f_0 = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{0.1 \cdot 10^{-3} \cdot 10 \cdot 10^{-6}}} $ Hz
  • Example 3:
    • In an LCR circuit, if the resonant frequency is 1000 Hz and the capacitance is 10 μF, find the inductance and resistance.
    • Solution:
      • Resonant frequency:
        • $ f_0 = \frac{1}{2\pi\sqrt{LC}} $
      • Substituting the given values:
        • $ 1000 = \frac{1}{2\pi\sqrt{L \cdot 10 \cdot 10^{-6}}} $
        • Solve for L.
  1. LC Oscillations - Examples
  • Example 1:
    • A series LCR circuit has an inductance of 0.2 H, a capacitance of 50 μF, and a resistance of 100 Ω.
    • Calculate the resonant frequency and the quality factor of the circuit.
    • Solution:
      • Resonant frequency: $ f_0 = \frac{1}{2\pi\sqrt{LC}} $
      • Quality factor: $ Q = \frac{1}{R} \sqrt{\frac{L}{C}} $
  • Example 2:
    • A capacitor of 10 μF and an inductor of 0.1 H are connected in series.
    • Determine the resonant frequency of the resulting circuit.
    • Solution:
      • Resonant frequency: $ f_0 = \frac{1}{2\pi\sqrt{LC}} $
  • Example 3:
    • In an LCR circuit, if the resonant frequency is 1000 Hz and the capacitance is 10 μF, find the inductance and resistance.
    • Solution:
      • Resonant frequency: $ f_0 = \frac{1}{2\pi\sqrt{LC}} $
  1. Lenz’s Law
  • Lenz’s law is a fundamental law in electromagnetic induction.
  • It states that when an induced current is produced by a changing magnetic field, the direction of the induced current opposes the change in the magnetic field.
  • This law is a consequence of the conservation of energy and Faraday’s law of electromagnetic induction.
  • Key Points:
    • The induced current generates a magnetic field that opposes the change in the magnetic field causing it.
    • This law is essential for understanding electromagnetic induction and the functioning of devices such as transformers and motors.
    • Lenz’s law also explains why eddy currents are generated in conductive materials when exposed to a changing magnetic field.
  1. Lenz’s Law - Equation
  • Lenz’s law can be mathematically expressed using the following equation: $ \text{Induced EMF} = -N \frac{d\phi}{dt} $ where:
    • Induced EMF is the electromotive force (voltage) induced in a circuit.
    • N is the number of turns in the coil or loop.
    • $ \frac{d\phi}{dt} $ is the rate of change of the magnetic flux.
  • The negative sign in the equation signifies that the induced current opposes the change in the magnetic field.
  1. Application of Lenz’s Law - Electric Generators
  • Lenz’s law plays a crucial role in the functioning of electric generators.
  • An electric generator converts mechanical energy into electrical energy.
  • When a coil of wire is rotated within a magnetic field, the magnetic flux through the coil changes, inducing an electromotive force (EMF).
  • According to Lenz’s law, the induced current in the coil generates a magnetic field that opposes the change in the original magnetic field, causing the coil to experience a torque.
  • This torque provides resistance to the rotation of the coil, and work is done to overcome this resistance, resulting in the conversion of mechanical energy to electrical energy.
  1. Application of Lenz’s Law - Transformers
  • Lenz’s law is also essential for understanding the operation of transformers.
  • A transformer consists of two coils, a primary coil and a secondary coil, coupled by a magnetic core.
  • When an alternating current flows through the primary coil, it creates an alternating magnetic field in the core.
  • The changing magnetic field induces an EMF in the secondary coil.
  • According to Lenz’s law, the induced current in the secondary coil produces a magnetic field that opposes the change in the original magnetic field.
  • This opposing magnetic field regulates the transfer of energy between the primary and secondary coils, allowing for efficient voltage transformation.
  1. AC versus DC - Comparison
  • AC (Alternating Current) and DC (Direct Current) are two different types of electric current.
  • AC current periodically changes its direction, while DC current flows in one direction continuously.
  • AC Current:
    • Alternates direction periodically, usually following a sine wave pattern.
    • Commonly used in power transmission and residential circuits.
    • Easily transformable to different voltage levels using transformers.
    • Can carry electricity over long distances with minimal loss.
  • DC Current:
    • Flows in one direction continuously.
    • Commonly used in electronic devices, batteries, and some specialized applications.
    • Requires conversion (rectification) from AC power sources for most applications.
    • Maintains constant voltage levels but is not as easily transformed as AC.
  1. Electromagnetic Waves - Overview
  • Electromagnetic waves are transverse waves composed of electric and magnetic fields.
  • They are generated by oscillating charged particles and can propagate through vacuum and various mediums.
  • Electromagnetic waves encompass a wide range of frequencies, forming the electromagnetic spectrum.
  • Electromagnetic Spectrum:
    • Consists of radio waves, microwaves, infrared waves, visible light, ultraviolet waves, X-rays, and gamma rays.
    • Each region of the spectrum has different properties and applications.
    • Visible light is the portion of the spectrum detectable by the human eye and is responsible for our perception of color.
  1. Electromagnetic Waves - Properties
  • Electromagnetic waves have several properties, including:
  1. Wavelength (λ):
    • The distance between two consecutive wave crests or troughs.
    • Expressed in meters (m) or other distance units.
  1. Frequency (f):
    • The number of wave cycles passing a point per unit time.
    • Expressed in hertz (Hz) or cycles per second (cps).
  1. Speed (v):
    • The speed at which an electromagnetic wave propagates through a medium.
    • In vacuum, all electromagnetic waves travel at the speed of light, denoted by “c” (~3.00 x 10^8 m/s).
  1. Amplitude (A):
    • The maximum displacement or magnitude of the electric or magnetic field in an electromagnetic wave.
  1. Polarization:
    • The orientation of the electric and magnetic fields of an electromagnetic wave relative to its direction of propagation.
  1. Electromagnetic Spectrum - Applications
  • The electromagnetic spectrum has various applications across different frequency ranges, such as:
  1. Radio Waves:
    • Used for communication, broadcasting, and navigation systems.
  1. Microwaves:
    • Used for cooking, satellite communication, and radar systems.
  1. Infrared Waves:
    • Used for remote controls, thermal imaging, and communication.
  1. Visible Light:
    • Used for vision, photography, optical communication, and color displays.
  1. Electromagnetic Spectrum - Applications (contd.)
  • Electromagnetic Spectrum - Applications (contd.)
  1. Ultraviolet Waves:
    • Used in sterilization, medical applications, and security purposes.
  1. X-rays:
    • Used in medical imaging, airport security, and scientific research.
  1. Gamma Rays:
    • Used in cancer treatment, industrial applications, and scientific research.
  • The wide range of applications highlights the importance of understanding and utilizing the properties and behaviors of electromagnetic waves throughout the electromagnetic spectrum.

21. Reflection of Light

  • Reflection is the bouncing back of light when it strikes a surface.
  • It follows two laws: the incident angle is equal to the angle of reflection, and the incident ray, reflected ray, and the normal to the surface all lie in the same plane.
  • Key points:
  • Reflection can occur from various surfaces, such as mirrors, water, or polished metal.
  • The angle between the incident ray and the normal is known as the angle of incidence.
  • The angle between the reflected ray and the normal is known as the angle of reflection.

22. Refraction of Light

  • Refraction is the bending of light as it passes from one medium to another.
  • It occurs due to the change in the speed of light as it moves through different media.
  • The change in direction is caused by the change in the velocity of light.
  • Key points:
  • The bending of light during refraction is governed by Snell’s Law, which relates the angle of incidence and the angle of refraction.
  • The refractive index of a medium is a measure of how much the light is bent when passing through it.

23. Snell’s Law

  • Snell’s Law describes the relationship between the angle of incidence and the angle of refraction of a light ray as it passes from one medium to another.
  • Mathematically, Snell’s Law can be written as: $ n_1 \sin\theta_1 = n_2 \sin\theta_2 $ , where:
  • $ n_1 $ and $ n_2 $ are the refractive indices of the first and second mediums, respectively.
  • $ \theta_1 $ and $ \theta_2 $ are the angles of incidence and refraction, respectively.
  • Key points:
  • Snell’s Law explains why light changes direction when it passes through interfaces between different materials.
  • It also determines the critical angle above which total internal reflection occurs.

24. Total Internal Reflection

  • Total internal reflection occurs when light traveling in a medium strikes the boundary of another medium at an angle greater than the critical angle.
  • In this case, all the light is reflected back into the original medium.
  • Key points:
  • Total internal reflection can only occur when light travels from a medium with a higher refractive index to a medium with a lower refractive index.
  • It is the principle behind optical fibers and several optical devices, such as prisms and binoculars.

25. Optical Fibers

  • Optical fibers are thin, flexible strands of highly transparent material used to transmit light signals over long distances.
  • They work based on the principle of total internal reflection.
  • Key points:
  • Optical fibers consist of a core (where the light is transmitted) surrounded by a cladding layer with a lower refractive index.
  • Light is transmitted through the core by repeated total internal reflections.
  • Optical fibers have revolutionized communication technology due to their ability to transmit large amounts of data with minimal loss.

26. Image Formation by Mirrors

  • Mirrors are used to form images by reflecting light.
  • The two most commonly used types of mirrors are plane mirrors and spherical mirrors.
  • Key points:
  • Plane mirrors form virtual images that are upright and laterally inverted.
  • Spherical mirrors can be either concave or convex.
  • Concave mirrors can form real or virtual images depending on the position of the object.
  • Convex mirrors always form virtual, upright, and diminished images.

27. Mirror Formula

  • The mirror formula is used to calculate the magnification, object distance, and image distance for mirrors.
  • Mathematically, the formula can be written as: $ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} $ , where:
  • f is the focal length of the mirror.
  • v is the image distance.
  • u is the object distance.
  • Key points:
  • The mirror formula is derived using the lens-maker’s formula.
  • It is valid for both convex and concave mirrors.

28. Lens Formula

  • The lens formula is used to calculate the magnification, object distance, and image distance for lenses.
  • Mathematically, the formula can be written as: $ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} $ , where:
  • f is the focal length of the lens.
  • v is the image distance.
  • u is the object distance.
  • Key points:
  • The lens formula is derived using the thin lens equation and Snell’s Law.
  • It is valid for both convex and concave lenses.

29. Lens Power

  • The power of a lens is a measure of its ability to converge or diverge light.
  • It is defined as the reciprocal of the focal length and is measured in diopters (D).
  • Key points:
  • The power of a lens can be positive or negative, depending on whether the lens is converging or diverging, respectively.
  • The power of a lens is related to its focal length by the equation: P = $ \frac{1}{f} $ .
  • The power of a lens can be used to determine the degree of refraction or the amount of magnification provided.

**30. Combined Lens and Mirror Systems

  • The combination of lenses and mirrors can be used to achieve various optical effects, such as image formation, focusing, and magnification.
  • Different combinations produce different effects and have unique optical properties.
  • Key points:
  • Compound lenses, made up of multiple lenses, can correct for various optical aberrations.
  • Reflecting telescopes use mirrors to gather and focus light, while refracting telescopes use lenses.
  • Understanding the behavior of combined lens and mirror systems is crucial in designing and analyzing optical devices and instruments.