Gravitational Fields: Physics Explained

• Gravitational fields and electric and magnetic fields are fundamental concepts in physics, describing how forces act at a distance.
• DC motors use the principles of electromagnetism to convert electrical energy into mechanical energy.
• Orbital motions are governed by gravitational forces, describing the movement of celestial bodies.
• Gravitational potential energy exists in both uniform and non-uniform gravitational fields.

Gravitational Fields

• Gravitational fields are regions of space surrounding a mass where another mass experiences a gravitational force.
• The gravitational field strength, g, is defined as the gravitational force per unit mass: g = F/m, measured in N/kg.
• Near the Earth's surface, the gravitational field is approximately uniform, with g ≈ 9.8 m/s² downwards.
• Gravitational field lines indicate the direction of the gravitational force; they point towards the center of the mass creating the field.
• For a point mass M, the gravitational field strength at a distance r from the mass is: g = GM/r², where G is the gravitational constant (6.674 × 10⁻¹¹ N⋅m²/kg²).

Gravitational Potential Energy in Uniform Fields

• Gravitational potential energy (GPE) in a uniform field is given by: ΔU = mgΔh, where m is the mass, g is the gravitational field strength, and Δh is the change in height.
• The reference point for zero GPE is arbitrary; usually, the ground or a convenient level is chosen.
• The work done against gravity to lift an object is equal to the change in its GPE.

Gravitational Potential Energy in Non-Uniform Fields

• In a non-uniform field, the gravitational force varies with distance.
• The gravitational potential energy (U) at a distance r from a point mass M is: U = -GMm/r.
• The zero point for GPE in a non-uniform field is defined as infinity (i.e., as r approaches infinity, U approaches 0).
• The negative sign indicates that GPE is always negative relative to infinity.
• The change in GPE when moving from r1 to r2 is: ΔU = -GMm/r2 - (-GMm/r1) = GMm(1/r1 - 1/r2).
• The escape velocity is the minimum velocity required for an object to escape the gravitational field of a planet: vesc = √(2GM/r).

Electric Fields

• Electric fields are regions of space surrounding an electric charge where another charge experiences an electric force.
• Electric field strength, E, is defined as the electric force per unit positive charge: E = F/q, measured in N/C or V/m.
• Electric field lines indicate the direction of the force on a positive test charge; they point away from positive charges and towards negative charges.
• For a point charge Q, the electric field strength at a distance r from the charge is: E = kQ/r², where k is Coulomb's constant (8.988 × 10⁹ N⋅m²/C²).
• The electric field inside a parallel plate capacitor is uniform: E = V/d, where V is the potential difference and d is the distance between the plates.

Magnetic Fields

• Magnetic fields are regions of space around a magnet or moving electric charge where a magnetic force is exerted.
• Magnetic field strength is represented by the symbol B, measured in Tesla (T).
• Magnetic field lines indicate the direction of the magnetic field; they form closed loops, emerging from the north pole and entering the south pole.
• A moving charge q with velocity v in a magnetic field B experiences a force: F = qvBsinθ, where θ is the angle between v and B.
• The magnetic force is perpendicular to both the velocity and the magnetic field direction (right-hand rule).
• A current-carrying wire in a magnetic field experiences a force: F = ILBsinθ, where I is the current, L is the length of the wire, and θ is the angle between the wire and the field.

DC Motors

• A DC motor converts electrical energy into mechanical energy using the principles of electromagnetism.
• A simple DC motor consists of a coil (armature) in a magnetic field, a commutator, and brushes.
• When current flows through the coil, it experiences a torque due to the magnetic force on the current-carrying wires.
• The torque is given by: τ = NBIA sinθ, where N is the number of turns, B is the magnetic field strength, I is the current, A is the area of the coil, and θ is the angle between the magnetic field and the normal to the coil.
• The commutator reverses the direction of current every half-rotation, ensuring continuous rotation of the coil.
• Back EMF (electromotive force) is generated in the coil as it rotates in the magnetic field, opposing the applied voltage: EMF = NBAω, where ω is the angular velocity.
• The net voltage in the circuit is: Vnet = Vapplied - EMF.

Orbital Motions

• Orbital motion is the motion of an object around another object due to gravity.
• Kepler's Laws of Planetary Motion describe the motion of planets around the Sun.
• Kepler's First Law: Planets move in elliptical orbits with the Sun at one focus.
• Kepler's Second Law: A line joining a planet and the Sun sweeps out equal areas during equal intervals of time (conservation of angular momentum).
• Kepler's Third Law: The square of the orbital period is proportional to the cube of the semi-major axis: T² ∝ r³, where T is the period and r is the semi-major axis.
• For circular orbits, the gravitational force provides the centripetal force: GMm/r² = mv²/r, where v is the orbital velocity.
• The orbital velocity is: v = √(GM/r).
• The orbital period is: T = 2πr/v = 2π√(r³/GM).
• Geostationary orbits are orbits where a satellite remains above the same point on Earth: T = 24 hours.