Topic: More Applications of Ampere’s Law - Toroid

Introduction

• Ampere’s law is a valuable tool for calculating the magnetic field created by certain symmetrical current distributions.
• In this lecture, we will explore more applications of Ampere’s law focusing on the toroid.

Toroid: Definition and Structure

• A toroid is a doughnut-shaped object with a hollow center.
• It consists of a long wire uniformly wound in multiple turns around the non-central region of the torus.
• The wire carries an electric current, producing a magnetic field within the toroid.
• The magnetic field lines are mostly confined within the toroid due to the circular path of the wire.

Ampere’s Law for a Toroid

• Ampere’s law states that the line integral of the magnetic field ( $\vec{B}$ ) along a closed path ( $\oint \vec{B} \cdot d\vec{l}$ ) is equal to the product of permeability of free space ( $\mu_0$ ) and the net current enclosed ( $I_{\text{enclosed}}$ ).
• For a toroid, we can calculate the magnetic field using a simple formula.
• The magnetic field ( $B$ ) inside a toroid with $N$ turns, carrying a current ( $I$ ), and having an average radius ( $R$ ) is given by: $B = \frac{{\mu_0 \cdot N \cdot I}}{{2 \pi \cdot R}}$

Magnetic Field Inside a Toroid

• The magnetic field inside a toroid is uniform and parallel to the axis of the toroid.
• This is due to the symmetrical arrangement of the current-carrying wire.
• The magnetic field lines are circular and concentric around the toroid’s axis.
• The magnetic field strength decreases with increasing radial distance from the axis.
• The direction of the magnetic field can be determined using the right-hand rule: by curling the fingers of the right hand in the direction of the current, the thumb points in the direction of the magnetic field inside the toroid.

Magnetic Field Outside a Toroid

• Unlike inside the toroid, the magnetic field outside is almost zero.
• This is because the magnetic field lines within the toroid form closed loops and do not extend outside.
• Thus, the external magnetic field due to a toroid is negligible.
• This property makes toroids suitable in applications where magnetic shielding is required.

Applications of Toroids

• Toroids are commonly used in electronics, telecommunications, and power distribution industries.
• Transformers: The toroidal shape allows for compact designs and efficient magnetic coupling.
• Inductors: Toroidal cores minimize magnetic field leakage and provide better inductance.
• Magnetic Shielding: Toroids are used to create magnetic shields around sensitive equipment, reducing the influence of external magnetic fields.

Example Problem 1

A toroid has 100 turns uniformly wound around it. The average radius of the toroid is 10 cm. If the current passing through the toroidal coil is 5 A, calculate the magnetic field inside the toroid. Given: Number of turns, N = 100 Average radius, R = 10 cm = 0.1 m Current, I = 5 A Using the formula: B = (μ₀⋅N⋅I) / (2⋅π⋅R) Substituting the values: B = (4π×10^{-7} × 100 × 5) / (2π×0.1) B ≈ 0.4 T

Example Problem 2

A toroid has 200 turns uniformly wound around it. The magnetic field inside the toroid is found to be 0.5 T. If the average radius of the toroid is 20 cm, calculate the current passing through the toroidal coil. Given: Number of turns, N = 200 Average radius, R = 20 cm = 0.2 m Magnetic field, B = 0.5 T Rearranging the formula: I = (2⋅π⋅R⋅B) / (μ₀⋅N) Substituting the values: I = (2π×0.2×0.5) / (4π×10^{-7} × 200) I ≈ 0.2 A

Summary

• A toroid is a doughnut-shaped object with a wire wound around it, carrying an electric current.
• Ampere’s law can be used to calculate the magnetic field inside a toroid using the formula B = (μ₀⋅N⋅I) / (2⋅π⋅R), where N is the number of turns, I is the current, and R is the average radius.
• The magnetic field inside a toroid is uniform and parallel to the axis, and almost zero outside.
• Toroids find applications in transformers, inductors, and magnetic shielding. Stay tuned for more exciting lectures on physics!

Magnetic Field Calculation

To calculate the magnetic field inside a toroid using Ampere’s law, follow these steps:

1. Determine the number of turns in the toroid (N).

2. Measure the average radius of the toroid (R).

3. Find out the current passing through the toroidal coil (I).

4. Use the formula B = (μ₀⋅N⋅I) / (2⋅π⋅R) to calculate the magnetic field (B).

5. Substitute the values into the formula and solve for B.

Magnetic Field Units

The units of magnetic field are derived from the units of other quantities involved in the formula.

• In SI units, the magnetic field is measured in tesla (T).
• 1 tesla is equivalent to 1 N/(A⋅m).
• Alternatively, the magnetic field can be expressed in gauss (G), where 1 T = 10,000 G.
• In CGS units, the magnetic field is measured in oersted (Oe).
• 1 oersted is equivalent to 79.5775 A/m.

Example Problem 3

A toroid with an average radius of 5 cm carries a current of 3 A. If it has 150 turns, calculate the magnetic field inside the toroid in mT. Given: Number of turns, N = 150 Average radius, R = 5 cm = 0.05 m Current, I = 3 A Using the formula: B = (μ₀⋅N⋅I) / (2⋅π⋅R) Substituting the values: B = (4π×10^{-7} × 150 × 3) / (2π×0.05) B ≈ 9.55 mT

Toroidal Inductor

• A toroidal inductor is a type of inductor that uses a toroid as its core.
• It is widely used in electrical circuits to store energy in the form of a magnetic field.
• The toroidal shape helps to minimize magnetic field leakage and improve inductance.
• Toroidal inductors are commonly used in high-frequency applications due to their low electrical resistance and small size.
• They are also known for their high Q factor (quality factor), which determines the efficiency of energy storage in the inductor.

Toroidal Transformer

• A toroidal transformer is a type of transformer that uses a toroid as its core.
• It is widely used in electronic devices and power distribution systems.
• The toroidal shape provides several advantages over traditional transformers, including compact size, low electromagnetic interference, and high efficiency.
• Toroidal transformers offer better magnetic coupling, resulting in improved voltage regulation and lower power losses.
• They are commonly found in audio amplifiers, computer power supplies, and other electronic equipment.

Comparison: Toroidal Transformer vs. E-Core Transformer

Toroidal Transformer:

• Efficient magnetic coupling
• Compact size and lighter weight
• Low electromagnetic interference
• Higher cost E-Core Transformer:
• Lower cost
• Higher magnetic field leakage
• Bigger in size and heavier
• Electromagnetic interference Choose the appropriate transformer based on the specific requirements and constraints of the application.

Magnetic Shielding

• Magnetic shielding refers to the ability to reduce the influence of external magnetic fields on a specific region.
• Toroids are often used to create magnetic shields around sensitive equipment or components.
• The toroid acts as a conductor for magnetic flux, diverting the external magnetic field lines away from the shielded area.
• This helps protect sensitive electronic circuits from interference and improves their performance.
• Magnetic shielding is crucial in applications such as MRI machines, particle accelerators, and electronic devices sensitive to magnetic fields.

Equation: Magnetic Flux Through a Toroid

The magnetic flux ( $\Phi_B$ ) through a toroidal coil is given by: $\Phi_B = B \cdot A \cdot N = \frac{{(\mu_0 \cdot N \cdot I) \cdot (\pi \cdot R^2)}}{{\mu_0}} = N \cdot I \cdot (\pi \cdot R^2)$ where B is the magnetic field, A is the cross-sectional area of the toroid, N is the number of turns, I is the current, and R is the average radius of the toroid.

Toroid vs. Solenoid

Toroid:

• Ring-shaped
• Magnetic field confined within the toroid
• Suitable for magnetic shielding
• Uniform magnetic field inside
• Difficult to adjust the magnetic field strength Solenoid:
• Cylindrical-shaped
• Magnetic field extends outside
• Used in electromagnets and speakers
• Non-uniform magnetic field inside
• Magnetic field strength can be adjusted by changing current or number of turns

Conclusion

• Ampere’s law can be applied to calculate the magnetic field inside a toroid.
• Toroids have various practical applications in electronics, telecommunications, and power distribution industries.
• Magnetic fields inside a toroid are uniform and parallel to the axis, while outside the toroid they are almost zero.
• Toroids are used in inductors, transformers, and magnetic shielding systems.
• Toroidal transformers and inductors offer advantages such as efficiency, compact size, and lower electromagnetic interference.

Physics Lecture: 12th Boards

Topic: Electromagnetic Waves

Introduction to Electromagnetic Waves

• Electromagnetic waves are transverse waves that consist of electric and magnetic fields oscillating perpendicular to each other.
• They are solutions to Maxwell’s equations and are characterized by their frequency ( $f$ ) and wavelength ( $\lambda$ ).
• Electromagnetic waves propagate through space and do not require a medium to travel.
• They range from long radio waves to short gamma rays, covering a wide spectrum of frequencies and wavelengths.
• Electromagnetic waves exhibit both wave-like and particle-like properties, known as wave-particle duality.

Electromagnetic Spectrum

The electromagnetic spectrum is the range of all possible frequencies of electromagnetic radiation.

• It includes various types of waves, such as radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.
• Each category of waves has different properties and applications.
• The spectrum is divided into regions based on frequency and wavelength, allowing us to categorize and study different electromagnetic waves.
• The speed of light ( $c$ ) is the same for all electromagnetic waves in a vacuum: $c ≈ 3 × 10^8$ m/s.

Properties of Electromagnetic Waves

1. Speed: Electromagnetic waves travel at the speed of light in a vacuum ( $c ≈ 3 × 10^8$ m/s).

2. Frequency ( $f$ ): The number of complete oscillations of the electric and magnetic fields per unit time (measured in hertz, Hz).

3. Wavelength ( $\lambda$ ): The distance between two consecutive wave crests or troughs (measured in meters, m).

4. Amplitude: The maximum extent of displacement from the equilibrium position.

5. Energy: Electromagnetic waves carry energy from one place to another.

Relationship between Frequency, Wavelength, and Speed

The relationship between frequency ( $f$ ), wavelength ( $\lambda$ ), and speed of light in a vacuum ( $c$ ) is given by the equation: $c = f \cdot \lambda$ where:

• $c$ is the speed of light ( $c ≈ 3 × 10^8$ m/s)
• $f$ is the frequency of the wave (measured in hertz, Hz)
• $\lambda$ is the wavelength of the wave (measured in meters, m) This relationship shows that as the frequency of an electromagnetic wave increases, its wavelength decreases, and vice versa.

Electromagnetic Wave Spectrum

The electromagnetic spectrum is categorized into the following regions based on frequency and wavelength:

1. Radio Waves:
   • Longest wavelengths and lowest frequencies.
   • Used in communication systems, such as radio and television broadcasting.

2. Microwaves:
   • Shorter wavelengths and higher frequencies than radio waves.
   • Used in microwave ovens, telecommunications, and radar systems.

3. Infrared (IR) Waves:
   • Wavelengths longer than visible light but shorter than microwaves.
   • Used in remote controls, thermal imaging, and heating applications.

4. Visible Light:
   • Wavelengths that are visible to the human eye.
   • Consists of different colors ranging from red (longest wavelength) to violet (shortest wavelength).

5. Ultraviolet (UV) Waves:
   • Wavelengths shorter than visible light.
   • Present in sunlight and used in sterilization, fluorescent lighting, and medical applications.

6. X-Rays:
   • Shorter wavelengths and higher frequencies than UV waves.
   • Used in medical imaging (X-rays), material testing, and security scanning.

7. Gamma Rays:
   • Smallest wavelengths and highest frequencies.
   • Produced by nuclear reactions and used in cancer treatment and sterilization.

Examples of Electromagnetic Waves

1. Radio Waves:
   • Used for AM/FM radio broadcasting, cell phones, and Wi-Fi.

2