Electric and Gravitational Fields

(a) Features of Electric and Gravitational Fields

Field: A region where a force is exerted on objects due to a physical quantity like mass or charge.

Electric Field:

Electric field strength, E, is the force per unit charge on a small positive test charge placed at the point

• Caused by electric charges.

• Can be attractive or repulsive (like charges repel, opposite charges attract).

Coulomb’s Law:

The force between two point charges is given by:

F = 9 × 10⁹ x Q₁Q₂ / r²

where 9 × 10⁹ Fm^-1 (k) is Coulomb’s constant, Q₁ and Q₂​ are charges, and r is the separation.

Electric Field Strength E:

The force per unit positive charge at a point:

E = F / q

Gravitational Field:

Gravitational field strength, g, is the force per unit mass on a small test mass placed at the point

• Caused by mass.

• Always attractive.

Newton’s Law of Universal Gravitation:

The force between two masses is given by:

F = G x M₁M₂ / r²

where G is the gravitational constant, M₁ and M₂​ are masses, and r is the separation.

Gravitational Field Strength g:

The force per unit mass at a point:

g = F / m

Inverse Square Law:

Both electric and gravitational forces decrease with the square of the distance (1/r²).

(b) Gravitational Field Outside a Spherical Body

Newton’s Shell Theorem:

Outside a uniform spherical mass (e.g., Earth), the gravitational field behaves as if all mass were concentrated at a single point at the centre.

Gravitational Field Strength Outside a Sphere:

g= GM / r²

where M is the total mass of the sphere and r is the radial distance from the centre.

(c) Field Lines (Lines of Force)

Field Lines:

Imaginary lines that show the direction of force in a field.

Electric Field Lines:

Point away from positive charges.

Point toward negative charges.

Gravitational Field Lines:

Always point toward the mass (gravity is always attractive).

Also radial for a point mass.

Density of Field Lines:

Indicates field strength (closer lines = stronger field).

(d) Equipotential Surfaces

Equipotential Surface:

A surface where the potential is the same at all points.

For a Point Charge or Mass:

Equipotential surfaces are spheres centred around the charge or mass.

Work Done on an Equipotential Surface:

Zero, since moving along the surface does not change potential energy.

Perpendicular to Field Lines:

• Electric and gravitational equipotentials are always at right angles to the respective field lines.

(e) Calculating Net Potential and Resultant Field Strength

Electric Potential V:

The electric potential at a point due to a charge q is:

V=kq / r

For multiple charges, add potentials algebraically (since potential is a scalar quantity).

Gravitational Potential Φ:

The gravitational potential at a point due to a mass M is:

Φ = −GM / r

• Always negative (gravity is always attractive).

• Also follows superposition (sum of individual potentials).

Field Strength (Vector Quantity):

• The net electric field at a point:

  E_net = ∑ Ei

• The net gravitational field at a point:

  g_net = ∑ gi​

Directions must be considered when summing vectors.

(f) Equation ΔUP=mgΔh Small Height Changes

Gravitational Potential Energy (U_P​) Change:

ΔU_P=mgΔh

where g is constant over small height changes (near Earth’s surface).

For Large Distance Changes:

When g varies significantly, use:

U_P = − GMm / r

instead of mgΔh