Electrostatic and gravitational fields of force

UNIT 4 Fields and Options 4.2 Electrostatic and gravitational fields of force Learners should be able to demonstrate and apply their knowledge and understanding of:
a)	The features of electric and gravitational fields as specified in the below table
b)	The idea that the gravitational field outside spherical bodies such as the Earth is essentially the same as if the whole mass were concentrated at the Centre
c)	Field lines (or lines of force) giving the direction of the field at a point, thus, for a positive point charge, the field lines are radially outward
d)	Equipotential surfaces joining points of equal potential and are therefore spherical for a point charge
e)	How to calculate the net potential and resultant field strength for a number of point charges or point masses
f)	The equation [math]\Delta U_P = mg\Delta h[/math] for distances over which the variation of g is negligible

a) The features of electric and gravitational fields

b) Gravitational Field Outside a Spherical Body
The gravitational field of a massive spherical body (like the Earth) behaves as if the entire mass were concentrated at a single point at its center. This concept is based on Newton’s Law of Universal Gravitation and the Shell Theorem.
1. Newton’s Law of Universal Gravitation
Newton’s law states that the force of gravity between two objects of masses M and m, separated by a distance r, is given by:
[math]F = \frac{GMm}{r^2}[/math]
Where:
G is the gravitational constant ([math]6.674 \times 10^{-11} \ \text{N} \text{m}^2 \text{kg}^{-2}[/math]),
M is the mass of the larger body (e.g., Earth),
m is the mass of a small object,
r is the distance between their centers of mass.
For a spherical body like Earth, this equation suggests that its gravitational effect on objects outside behaves as if all the mass were concentrated at the center.
2. Shell Theorem Explanation
The Shell Theorem, formulated by Newton, states:
Outside a uniform spherical body: The gravitational field behaves as if all the mass were concentrated at a single point at the center.
Figure 1 Newton’s shell theorem
Inside a uniform spherical shell: The gravitational field inside a hollow sphere is zero.
Thus, for objects outside the Earth, gravity follows an inverse-square law similar to that of a point mass.
Example:
If an astronaut is 1000 km above Earth’s surface, the gravitational attraction can be calculated as if all of Earth’s mass were concentrated at the center.
The field strength decreases with distance, following [math]g = \frac{GM}{r^2}[/math].
c) Field Lines & Their Direction
A field line (or line of force) represents the direction in which a force acts on a small test object in a field.
Characteristics of Field Lines:
They indicate the direction of force at any point in space.
Field strength is proportional to line density (closer lines = stronger field).
Field lines never cross because the force at a point has only one direction.
Figure 2 Field lines and their direction
1. Gravitational Field Lines
Gravitational field lines point toward a massive body, as gravity is always attractive.
The lines are radially inward for a spherical object like Earth.
Example:
Around Earth, field lines start from infinity and converge at the center.
The density of these lines decreases as distance increases, showing that gravity weakens with distance.
2. Electric Field Lines (for a Positive Charge)
For a positive point charge, the electric field lines radiate outward because a positive charge repels other positive charges.
For a negative point charge, the field lines are radially inward because a negative charge attracts positive charges.
Example:
The Sun’s gravitational field is similar to an electric field around a negative charge, pulling objects inward (e.g., planets).
The field around a proton resembles that of a positive point charge, pushing away other protons.
Comparison: Gravitational vs. Electric Field Lines

Feature	Gravitational Field Lines	Electric Field Lines
Force Type	Always attractive	Can be attractive or repulsive
Direction	Toward mass (e.g., Earth)	Radial outward for positive, inward for negative
Field Strength	Decreases with [math]\frac{1}{r^2}[/math]	Decreases with [math]\frac{1}{r^2}[/math]
Example	Earth pulling objects downward	Proton repelling other protons

Conclusion
Gravitational fields of spherical bodies behave as if all mass were at the center, following the inverse-square law.
Field lines show the direction of force, with gravitational lines pointing inward and electric lines outward for positive charges.
d) Equipotential Surfaces
An equipotential surface is a surface where the electric potential (or gravitational potential) is the same at every point.
Figure 3 Equipotential surface
Properties of Equipotential Surfaces
No Work is Done Moving Along an Equipotential Surface – Since potential is the same at all points, moving a charge or mass along it requires zero work.
Always Perpendicular to Field Lines – The electric or gravitational field is always at right angles to the equipotential surfaces.
Closer Equipotential Surfaces = Stronger Field – Where the field is strong, equipotential surfaces are closer together.
Equipotential Surfaces for Different Fields
1️ For a Point Charge (Electric Field):
The equipotential surfaces are spherical shells centered on the charge.
The potential decreases radially outward.
Similar to the case of a gravitational field around a point mass.
2️ For a Uniform Electric Field:
Equipotential surfaces are parallel planes perpendicular to the field.
Example: The field between two parallel plates.
3️ For Gravitational Fields Around a Planet:
Equipotential surfaces around a spherical mass (e.g., Earth) are concentric spheres.
In uniform fields (e.g., near Earth’s surface), equipotential surfaces are horizontal planes.
Example:
Near Earth’s surface, the gravitational equipotential surfaces are horizontal planes (constant height).
Around a planet, equipotential surfaces form concentric spheres.
e) Net Potential and Resultant Field Strength for Multiple Charges or Masses
Electric Potential Due to Multiple Point Charges
The electric potential at a point due to multiple charges is given by:
[math]V_{\text{net}} = V_1 + V_2 + V_3 + \dots[/math]
Where:
[math]V = k \frac{Q}{r}[/math]
k is Coulomb’s constant ([math]9.0 \times 10^9 \,\text{Nm}^2/\text{C}^2[/math]).
Q is the charge.
r is the distance from the charge to the point.
Potential is a scalar quantity, so potentials from multiple charges are simply added algebraically (with sign).
Example:
For two charges +Q and −Q separated by a distance d, the potential at a point midway between them is:
[math]V_{\text{net}} = \frac{kQ}{r} + \frac{k(-Q)}{r} \\
V_{\text{net}} = 0[/math]
Figure 4 Electric potential due to multiple point charges
Gravitational Potential for Multiple Masses
Similar to electric potential, the gravitational potential due to multiple masses is:
[math]V_{\text{net}} = V_1 + V_2 + V_3 + \dots[/math]
Where:
[math]V = – \frac{GM}{r}[/math]
G is the gravitational constant ([math]6.674 \times 10^{-11} \,\text{Nm}^2/\text{kg}^2[/math]).
M is the mass.
r is the distance from the mass to the point.
Gravitational potential is always negative (attractive force).
Example:
For two masses [math]M_1[/math] and [math]M_2[/math] the net gravitational potential at a point is:
[math]V_{\text{net}} = -\frac{G M_1}{r_1} – \frac{G M_2}{r_2}[/math]
Resultant Field Strength (Electric & Gravitational Fields)
Since electric and gravitational fields are vector quantities, we must use vector addition to find the net field.
1️ Electric Field Due to Multiple Charges:
[math]E_{\text{net}} = E_1 + E_2 + E_3 + \dots[/math]
Direction matters: The electric field is directed away from positive charges and toward negative charges.
2️ Gravitational Field Due to Multiple Masses:
[math]g_{\text{net}} = g_1 + g_2 + g_3 + \dots[/math]
Gravitational field lines always point toward mass.
Use vector addition for multiple masses.
Example:
For two equal masses M separated by a distance, the net gravitational field strength at a point along the line joining them is:
[math]g_{\text{net}} = g_1 – g_2[/math]
(if in opposite directions).
f) The Equation [math]\Delta U_p = mg \Delta h[/math] for Small Height Changes
Gravitational Potential Energy Change
For small changes in height near Earth’s surface, where the gravitational field is nearly constant, the change in gravitational potential energy is:
[math]\Delta U_p = mg \Delta h[/math]
Where:
[math]\Delta U_p[/math] = Change in gravitational potential energy (J).
m = Mass of the object (kg).
g = Acceleration due to gravity (m/s²).
Δh = Change in height (m).
When height change is small compared to Earth’s radius ([math]R_E[/math])
When gravitational acceleration g remains nearly constant
For everyday height changes (e.g., climbing stairs, jumping).
For Large Height Changes: Use the Full Gravitational Potential Energy Formula
If h is very large, g is no longer constant, and we must use:
[math]U = -\frac{G M m}{r}[/math]
Where [math]r = R_E + h[/math].
Example Calculations
Example 1: Lifting an Object
A 5 kg object is lifted 3 meters above the ground. What is the change in gravitational potential energy?
[math]\Delta U_p = mg \Delta h \\
\Delta U_p = (5)(9.8)(3) \\
\Delta U_p = 147 \text{ J}[/math]
Example 2: Potential at a Distance
Find the gravitational potential at 1000 km above Earth where[math]R_E = 6.37 \times 10^6 \,\text{m}.[/math]
[math]V = -\frac{G M}{r} \\
V = -\frac{(6.674 \times 10^{-11}) (5.972 \times 10^{24})}{(6.37 \times 10^6 \times 10^6)}[/math]
Solving gives [math]V \approx -5.59 \times 10^7 \,\text{J/kg}.[/math].