A-Level Physics: Advanced Dynamics & Fields

Front

What is the derivation for the escape velocity from a planet using energy conservation?

Back

Start with the conservation of energy equation: Initial Total Energy (KE + GPE) = Final Total Energy (0 at infinity). So, 0.5mv² - (GMm)/r = 0. Solving for v yields v = √(2GM/r). Substituting g = GM/r² gives the alternate form v = √(2gr).

Front

How does the period of a satellite relate to the orbital radius according to Kepler's Third Law?

Back

The period T is related to the orbit radius r by T² ∝ r³. Derivation: Equating gravitational force to centripetal force (GMm/r² = mrω²) and using ω = 2π/T leads to T² = (4π²/GM) × r³. This applies to circular orbits assuming M >> m.

Front

Define Gravitational Potential (φ) and explain its scalar nature at infinity.

Back

Gravitational potential is the work done per unit mass in bringing a small test mass from infinity to a point in the field. Formula: φ = -GM/r. It is a scalar quantity (measured in J kg⁻¹) and is defined as zero at infinity. The negative value indicates a bound system requiring energy to escape.

Front

What is the relationship between Electric Field Strength (E) and Electric Potential (V)?

Back

Electric field strength is the potential gradient: E = -dV/dr. For a radial field, this becomes E = -ΔV/Δr. Since V = Q/(4πε₀r), differentiating with respect to r gives E = Q/(4πε₀r²). The negative sign indicates the field points in the direction of decreasing potential.

Front

Describe the discharge behavior of a capacitor through a resistor.

Back

The charge Q decays exponentially according to Q = Q₀e^(-t/RC). The current I follows I = I₀e^(-t/RC), and the voltage across the capacitor is V = V₀e^(-t/RC). The time constant τ (tau) is the time taken for the charge, current, or voltage to fall to 37% (1/e) of its initial value.