Gravitational Fields
What is a Gravitational Field?
A gravitational field is a region of space where a mass experiences a gravitational force. All objects with mass create a gravitational field around them.
Gravitational field strength g is defined as the force per unit mass at a point in the field.
For a point mass or outside a uniform sphere:
where G = 6.67 × 10−11 N m² kg−2, M is the mass creating the field, and r is the distance from the centre of the mass.
Radial Gravitational Field
Around a spherical mass like the Earth, the gravitational field is radial — field lines point inward towards the centre of mass.
Radial gravitational field around Earth — field lines point inward towards the centre
Worked Example
Calculate the gravitational field strength at a height of 400 km above the Earth's surface. (RE = 6.4 × 106 m, ME = 6.0 × 1024 kg)
Step 1: r = RE + h = 6.4 × 106 + 400 × 103 = 6.8 × 106 m
Step 2: g = GM/r² = (6.67 × 10−11 × 6.0 × 1024) / (6.8 × 106)²
Step 3: g = 4.0 × 1014 / 4.624 × 1013 = 8.7 N kg−1
Key Facts
- Gravitational fields are always attractive — masses always attract each other
- Field strength follows an inverse-square law: doubling the distance quarters the field strength
- Near Earth's surface, g ≈ 9.81 N kg−1 and is approximately uniform
Exam Tip
Always check whether the question gives the distance from the surface or from the centre of the Earth. You need the distance from the centre for g = GM/r².
Gravitational Potential
Gravitational Potential (V)
The gravitational potential at a point is the work done per unit mass in bringing a test mass from infinity to that point.
Gravitational potential is always negative (zero at infinity) because work must be done against gravity to move a mass away.
Orbits & Kepler's Third Law
For a circular orbit, gravitational force provides the centripetal force:
Kepler's third law relates orbital period T to orbital radius r:
Worked Example
Calculate the orbital period of a satellite at height 400 km above the Earth.
Step 1: r = 6.4 × 106 + 4.0 × 105 = 6.8 × 106 m
Step 2: T² = (4π²/GM) r³ = (4π² / (6.67 × 10−11 × 6.0 × 1024)) × (6.8 × 106)³
Step 3: T² = (9.87 × 10−14) × 3.14 × 1020 = 3.10 × 107
Step 4: T = 5570 s ≈ 93 minutes
Key Facts
- Gravitational potential energy: Ep = mV = −GMm/r
- Escape velocity: vesc = √(2GM/r)
- Field strength is the negative potential gradient: g = −dV/dr
Electric Fields
Electric Field Strength
An electric field is a region where a charged particle experiences a force. Electric field strength E is the force per unit positive charge:
Uniform Electric Field
Between parallel plates with potential difference V and separation d:
Uniform electric field between parallel plates — equally spaced field lines from + to −
Radial Electric Field & Coulomb's Law
For a point charge or spherical charge Q, the field is radial:
Coulomb's law gives the force between two point charges:
where ε0 = 8.85 × 10−12 F m−1. The constant 1/(4πε0) ≈ 9.0 × 109 N m² C−2.
Worked Example
Calculate the force between charges of +3.0 μC and −5.0 μC separated by 0.10 m.
Step 1: F = Q1Q2 / (4πε0r²)
Step 2: F = (9.0 × 109) × (3.0 × 10−6 × 5.0 × 10−6) / (0.10)²
Step 3: F = 9.0 × 109 × 1.5 × 10−11 / 0.01 = 13.5 N (attractive)
Exam Tip
Electric fields can be attractive OR repulsive (unlike gravity which is always attractive). Always state the direction and nature of the force in your answer.
Electric Potential
Electric Potential (V)
The electric potential at a point is the work done per unit positive charge in bringing a small positive test charge from infinity to that point:
Unlike gravitational potential, electric potential can be positive (near positive charges) or negative (near negative charges).
Comparing Gravitational & Electric Fields
| Property | Gravitational | Electric |
|---|---|---|
| Source | Mass | Charge |
| Force type | Always attractive | Attractive or repulsive |
| Field strength | g = GM/r² | E = Q/(4πε0r²) |
| Potential | V = −GM/r (always −ve) | V = Q/(4πε0r) (+ve or −ve) |
Key Facts
- Both gravitational and electric fields obey inverse-square laws
- Field strength equals the negative potential gradient: E = −dV/dr
- Equipotential surfaces are perpendicular to field lines
Capacitance
Capacitance & Energy Storage
Capacitance is the charge stored per unit potential difference:
Energy stored in a capacitor:
Worked Example
A 470 μF capacitor is charged to 12 V. Calculate the energy stored.
Step 1: E = ½CV²
Step 2: E = 0.5 × 470 × 10−6 × 12²
Step 3: E = 0.5 × 470 × 10−6 × 144 = 0.034 J
Charging & Discharging
The time constant τ = RC determines how quickly a capacitor charges or discharges.
Discharging: Q = Q0e−t/RC V = V0e−t/RC I = I0e−t/RC
After one time constant, the value falls to 1/e ≈ 37% of its initial value.
Capacitor charging and discharging curves — exponential approach to final value
Exam Tip
To find the time constant from a graph, read off the time at which the value has fallen to 37% of its initial value (discharging) or risen to 63% of its final value (charging).
Magnetic Fields
Magnetic Flux & Flux Density
Magnetic flux density B (tesla, T) is the force per unit length per unit current on a conductor perpendicular to the field.
Magnetic flux Φ through an area A:
Flux linkage for a coil of N turns:
Force on a Current-Carrying Conductor
A wire of length L carrying current I in a magnetic field B experiences a force:
Direction given by Fleming's left-hand rule (thuMb = Motion, First finger = Field, seCond finger = Current).
Force on a Moving Charge
A charge Q moving at velocity v perpendicular to a magnetic field B experiences:
This force is always perpendicular to the velocity, causing circular motion. The radius of the circular path is r = mv/(BQ).
Worked Example
A wire of length 0.30 m carries a current of 4.0 A perpendicular to a field of 0.50 T. Find the force.
F = BIL sinθ = 0.50 × 4.0 × 0.30 × sin 90°
F = 0.60 N
Transformers
A transformer changes alternating voltage using electromagnetic induction between two coils on a shared iron core:
For an ideal transformer: VsIs = VpIp (power is conserved).
Key Facts
- 1 tesla = 1 Wb m−2
- A charged particle moving parallel to a magnetic field experiences no force
- Step-up transformers increase voltage (Ns > Np); step-down transformers decrease it
Electromagnetic Induction
Faraday's Law
The magnitude of the induced EMF is equal to the rate of change of magnetic flux linkage:
The negative sign reflects Lenz's law: the induced EMF acts in a direction to oppose the change in flux that produces it.
Lenz's Law
The direction of the induced current is such that it opposes the change producing it. This is a consequence of conservation of energy.
Worked Example
The flux through a 200-turn coil changes from 0.05 Wb to 0.01 Wb in 0.02 s. Find the induced EMF.
Step 1: ΔΦ = 0.05 − 0.01 = 0.04 Wb
Step 2: ε = N × ΔΦ/Δt = 200 × 0.04 / 0.02
Step 3: ε = 400 V
Key Facts
- An EMF is induced whenever the flux linkage through a circuit changes
- Moving a magnet into a coil induces a current; moving it out induces a current in the opposite direction
- Faster motion or more turns produces a larger EMF
Exam Tip
Lenz's law questions are common. Remember: the induced effect always opposes the cause. If flux is increasing, the induced current creates a field to reduce the flux.
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