Electromagnetism: Key Concepts and Principles
Electromagnetism
Kepler’s Second Law
The radius vector of each planet sweeps out equal areas in equal times. This also implies that the sector velocity of each planet is constant.
Universal Gravitation
The gravitational interaction between two bodies can be expressed as a single central attractive force proportional to the masses and inversely proportional to the square of the distance separating them.
Gravitational Field
A mass, M, creates a vector quantity known as gravitational field strength, g, at a point in space, P. This is defined as the force per unit mass placed at P.
Field Overlap
The resulting field at a point P is the sum of the fields that each body would create at that point if it were alone.
Work Done by a Force
The work done by a force (F) on a body depends on the magnitude of the force, the path length, and the relative orientation of the force vector with respect to the displacement. Only in the case of a constant force (i.e., not dependent on the position) can the work be determined by W = F * Δr * cos(x).
Work-Kinetic Energy Theorem
The work done on a particle by all the forces acting on it is equal to the change in its kinetic energy.
Conservative Force
A force is conservative when the work done by it, acting on a body between two points, is independent of the path taken.
Physical Meaning of Potential Energy at a Point A
The potential energy of a mass, m, at a point, A, in a gravitational field coincides with the work performed when “m” moves from A to a point at infinity.
Relationship Between Field and Potential
The physical meaning of potential is that of potential energy per unit mass.
Energy Conservation Principle
The total mechanical energy (kinetic + potential) of a particle moving under the influence of conservative forces remains constant.
Electric Charges: Features
- There are two types of electric charges (positive and negative).
- Electric charge is a scalar quantity.
- Electric charge follows the principle of additivity (charges add up).
- Electric charge exhibits complementarity (neutral matter has equal amounts of positive and negative charges).
- The magnitude of electric charge is quantized.
- Electric charge is conserved in closed systems.
- The unit of electric charge is the coulomb (C).
Coulomb’s Law
The force of attraction or repulsion between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance separating them. The forces are equal in magnitude and opposite in direction. The force can be expressed mathematically as:
F = k * (q1 * q2) / r²
where:
- F is the force between the charges
- k is Coulomb’s constant (approximately 9 x 10⁹ N⋅m²/C²)
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges
It is important to consider the unit vector directed from the charge exerting the force to the charge experiencing the force, and the charges’ signs must be included in the calculation.
Electrostatic Force: Characteristics
- It is a central force (acts along the line connecting the charges).
- It is a conservative force.
- It can be attractive or repulsive.
- It depends on the surrounding medium.
- It is a long-range force.
Electric Field (E)
The electric field at a point is defined as the force per unit positive charge placed at that point:
E = F / q
If we substitute Coulomb’s law for F, we obtain the electric field created by a point charge:
E = k * q / r²
Field Lines
Field lines are lines tangent to the electric field vector at each point. They provide a visual representation of the electric field.
Relationship Between Electric Field and Potential
The electric field is related to the electric potential (V) by the following equation:
E = -∇V
where ∇ is the gradient operator.
Force on a Charge in an Electric Field
The force (F) on a charge (q) in an electric field (E) is given by:
F = qE
Meaning of Potential Energy of a Charge q at a Point
The potential energy of a charge q at a point in an electric field is the work done by the field when a unit positive charge moves from that point to infinity.
U = qV
Meaning of Electric Potential at a Point
The electric potential at a point in an electric field is the work done by the field in moving a unit positive charge from a reference point (usually infinity) to that point.
Expression for Potential Energy of a Charge q’ in the Field Created by q
U = k * (q * q’) / r
Relationship Between Potential Energy of a Charge and Potential at that Point
U = qV
Electrically Conductive Materials: Characteristics
- They possess free charge carriers.
- In the absence of an applied electric field, charge carriers have random motion.
- When an external electric field is applied, the carriers respond with an orderly movement (electric current).
- A conductor is in electrostatic equilibrium when there is no orderly movement of charge inside and the following conditions are met:
- The electric field inside is zero.
- All points within the conductor have the same potential.
- The electric field vector is perpendicular to the surface.
- If the conductor is charged, the charges accumulate on the surface.
Dielectric or Insulating Materials: Characteristics
- They do not have free charge carriers.
- Under the action of external electric fields, they can polarize.
- Polarization is a displacement of charge within each molecule of the material.
- Dipoles are formed in the presence of external fields, resulting in charge densities on the surfaces.
- These surface charges create an internal electric field that opposes the external field, effectively reducing the net field inside the material.
Mechanical Energy of a Moving Charge in an Electric Field
The mechanical energy (kinetic + potential) of a charge (q) moving in an electric field is given by:
1/2 * m * v² + qV = constant
where:
- m is the mass of the charge
- v is the velocity of the charge
- V is the electric potential at the charge’s location
Oersted’s Experiment
Oersted’s experiment demonstrated the relationship between electricity and magnetism. He observed that an electric current deflected a compass needle, indicating that electric currents produce magnetic fields.
Origin of Magnetic Forces
Magnetic forces originate from the interaction between moving charges.
Complementarity of Electric and Magnetic Effects
Electric and magnetic effects are complementary and depend on the observer’s frame of reference. A charge that is stationary for one observer may be in motion for another, and thus experience a magnetic force.
Magnetic Field Created by a Rectilinear Current
B = (μ₀ * I) / (2π * r)
where:
- B is the magnetic field strength
- μ₀ is the permeability of free space (4π x 10⁻⁷ T⋅m/A)
- I is the current
- r is the distance from the wire
Magnetic Field at the Center of a Current Loop
B = (μ₀ * I) / (2 * r)
where r is the radius of the loop.
Magnetic Field Inside a Solenoid
B = μ₀ * n * I
where n is the number of turns per unit length.
Force on a Moving Charge (Lorentz Law)
If a charge q moves with velocity v within a magnetic field B, it experiences a force F with the following characteristics:
- It is proportional to the charge q.
- It is proportional to the velocity v and the magnetic field B.
- It is perpendicular to the plane determined by v and B.
F = q * v x B
Why Magnetic Forces Do No Work
Magnetic forces do no work because they are always perpendicular to the velocity of the charge.
Deduction of Expressions for Circular Motion in a Magnetic Field
For a charge moving in a uniform magnetic field, the magnetic force provides the centripetal force, resulting in circular motion. Equating the magnetic force (qvB) to the centripetal force (mv²/R), we can derive expressions for the radius of the circular path (R), the period of revolution (T), and the frequency of rotation (f):
- R = mv / qB
- T = 2πR / v = 2πm / qB
- f = 1 / T = qB / 2πm
Magnetic Force on a Straight Current-Carrying Wire
- It is proportional to the current I.
- It is proportional to the length of the conductor and the magnetic field B.
- It is perpendicular to the plane determined by the conductor and the magnetic field.
F = I * L x B
Force Between Parallel Currents
- The magnetic field created by current I₁ at a distance d is B₁ = (μ₀ * I₁) / (2π * d).
- The force exerted by this magnetic field on a parallel current I₂ is F₁₂ = I₂ * L * B₁.
- The force on current I₁ due to current I₂ is of the same magnitude but opposite in direction.
- The force per unit length between parallel currents is F/L = (μ₀ * I₁ * I₂) / (2π * d).
- Currents in the same direction attract each other, while currents in opposite directions repel each other.
Ampere (A)
One ampere is the current intensity flowing through two infinitely long, straight, parallel conductors, separated by a distance of 1 meter in a vacuum, that produces a force of 2 x 10⁻⁷ N per meter of length between them.
Coulomb (C)
One coulomb is the amount of charge that passes through a cross-section of a conductor in one second when the current is one ampere.
Magnetic Flux Through a Surface
The magnetic flux (Φ) is a measure of the number of magnetic field lines passing through a given surface.
Φ = B * A * cos(θ)
where:
- B is the magnetic field strength
- A is the area of the surface
- θ is the angle between the magnetic field and the surface normal
Magnetic Flux Through a Closed Surface
Since magnetic field lines are closed loops, the net magnetic flux through a closed surface is always zero.
Electromagnetic Induction
Electromagnetic induction is the process of generating an electric current from a changing magnetic field.
Faraday-Lenz Law
Faraday’s law of induction states that the induced electromotive force (emf) in a closed loop is equal to the negative rate of change of magnetic flux through the loop.
emf = -dΦ/dt
Lenz’s law states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it.
Conditions for Inducing a Current
- Connecting or disconnecting the primary circuit of a transformer (but not when the current is constant).
- Relative motion between a coil and a magnetic field.
- Moving a magnet relative to a coil.
Meaning of the Negative Sign in Faraday’s Law
The negative sign indicates that the induced emf and current oppose the change in magnetic flux.
Is the Magnetic Force Conservative?
No, the magnetic force is not conservative because it cannot be associated with a potential energy function.
Producing Alternating Current (AC)
An alternating current is a current that periodically reverses its direction. The phenomenon of electromagnetic induction allows us to build AC generators. If the magnetic flux varies sinusoidally, the induced emf and current will also be sinusoidal.
Transformer
Transformers are electrical devices based on electromagnetic induction that can change the voltage of an alternating current. They consist of two coils wound around a common ferromagnetic core. When an AC voltage is applied to the primary coil, it creates a changing magnetic flux in the core, which in turn induces an AC voltage in the secondary coil. The ratio of the number of turns in the primary and secondary coils determines the voltage transformation ratio.
Transformer Voltage Ratio
V₁ / V₂ = N₁ / N₂
where:
- V₁ and V₂ are the primary and secondary voltages, respectively
- N₁ and N₂ are the number of turns in the primary and secondary coils, respectively
If N₁ > N₂, the transformer is a step-down transformer (it decreases the voltage), and if N₁ < N₂, it is a step-up transformer (it increases the voltage).
Advantages of Using Alternating Currents
- Over 90% of the electricity consumed worldwide is AC.
- AC motors are simpler and more reliable than DC motors.
- AC can be easily stepped up or down in voltage using transformers, which reduces transmission losses.
- AC can be readily rectified (converted to DC) if needed.
Dielectric or Insulating Materials: Characteristics (repeated)
- They do not have free charge carriers.
- Under the action of external electric fields, they can polarize.
- Polarization is a displacement of charge within each molecule of the material.
- Dipoles are formed in the presence of external fields, resulting in charge densities on the surfaces.
- These surface charges create an internal electric field that opposes the external field, effectively reducing the net field inside the material.
Mechanical Energy of a Moving Charge in an Electric Field (repeated)
1/2 * m * v² + qV = constant
Oersted’s Experiment (repeated)
Oersted’s experiment revealed the relationship between electricity and magnetism. He observed that an electric current deflected a compass needle, indicating that electric currents produce magnetic fields.