Fields

1. Force field as a region:

• A force field is a region in space where a body experiences a non-contact force. It’s a fundamental concept in physics that describes the interaction between objects without physical contact.
• Types of force fields:
  1. Gravitational force field: surrounds massive objects, like planets and stars, and causes attraction.
  2. Electric force field: surrounds charged particles and objects, and causes attraction or repulsion.
  3. Magnetic force field: surrounds magnets and current-carrying wires, and causes attraction or repulsion.
  4. Radiative force field: associated with electromagnetic waves, like light and radiation.
• 
  Figure 1 non-contact forces
• Characteristics:
  – Non-contact: forces act without physical touch
  – Vector field: force fields have magnitude and direction
  – Spatial extent: force fields occupy a region in space
  – Intensity: force fields can vary in strength
• Force fields are a powerful tool for understanding and predicting the behavior of objects in various physical situations, from gravity to electromagnetism.

2. A force field as a vector:

• A force field can be represented as a vector, which is a mathematical object with both magnitude (size) and direction.
• The vector representation of a force field is a powerful tool for analyzing and visualizing the forces that act on objects within the field.

• ⇒ Electric field strength:

• A charge produces an electric field around it, which exerts a force on another charged object.

  
  Figure 2 Electric field strength

• This idea is similar to a magnetic field close to a magnet, or a gravitational field around a planet.
• An electric field strength is defined by the equation.
• [math] E = \frac {F}{Q} [/math]

• In a uniform field

• [math] E = \frac {V}{d} [/math]

• And generally
• [math] E = -\frac {\Delta V}{\Delta r} [/math]
• The E-r and V-r graphs below show the relation clearly.
• The gradient of the V-r graph is negative. So, the negative of its gradient gives a positive value for E in the E-r graph.
• 
  Figure 3 E-r and V-r graph
• Where F is the force, in newtons, which acts on a charge, Q, in coulombs.
• So electric field strength is measured in newtons per coulomb, [math] NC^{-1} \, \text{or} \, Vm^{-1}[/math] .
• The direction of the electric field is defined as the direction of the force on a positive charge.
• Electric field is a vector quantity because it has both magnitude and direction.
• The fields are uniform because in all places the field has the same strength and the same direction.
• The field lines start on a positive charge and end on a negative charge.

⇒Examples for electric field strength:

(1)
Force on a small charge
A small charge of +2µC is placed in the electric field in figure 4. What force does it experience?

Figure 4 Direction of electric field for 400NC^-1

Solution:

[math]
E = \frac{F}{Q}\\
F = EQ \\
F = 400 \times 2 \times 10^{-6}\\
F = 8 \times 10^{-4} \, \text{N} [/math]

(2)

Change in electric potential

Use figure 5 to calculate the change in potential in moving from position r=0 to r=0.2 m.

Figure 5 generally electric field in graph

Solution:

[math]
\Delta V = \text{Area under the graph} \\
\Delta V = \frac{1}{2} (E_2 + E_1) \cdot \Delta r \\
\Delta V = \frac{1}{2} (2000 + 1500) \cdot 0.2 \\
\Delta V = \frac{1}{2} \cdot 3500 \cdot 0.2 \\
\Delta V = 1750 \cdot \frac{2}{10}\\
\Delta V = 350 \, \text{V}[/math]

(3)

Calculate the electric field when a power supply applied potential difference of 220V and distance applied is 0.6m.

Solution:

The electric field strength

[math] E = \frac{V}{d}\\
E = \frac{220}{0.6}\\
E = 367 \, \text{V} \cdot \text{m}^{-1}[/math]

3. Force fields arise from the interaction:

• Force fields can arise from the interaction of:
  1. Mass: Gravitational force fields are generated by the interaction of masses, such as planets, stars, and galaxies.
     • An attractive force that exists between all objects with mass; an object with mass attracts another object with mass; the magnitude of the force is directly proportional to the masses of the two objects and inversely proportional to the square of the distance between the two objects.

      

     Figure 6 Gravitational field force

  2. Static charge: Electric force fields are generated by the interaction of static charges, such as positive and negative charges on particles or objects.
     • Electrostatic forces arise from the charge of two interacting surfaces, which can be two solid surfaces, but can also relate to a solid surface and a bacterium.

      

     Figure 7 static and moving charges

     • Static electricity involves charged objects that are static, which means not moving.
     • Static charged objects create an electric field that interacts with other charged objects around it.
     • When there is a consistent supply of electrons (negative terminal) and another area for those electrons to flow (positive terminal) you get a current.
     • A current is a flow of electrons.
     • They don’t flow as directly as our animation here but that is a lesson for a future unit. While a static charge creates a electric field a moving charge or current creates a magnetic field around it.
     • Static (Non-moving) charges creates an electric field.
     • Current (Moving) charges create a magnetic field.

  3. Moving charges: Magnetic force fields are generated by the interaction of moving charges, such as electric currents in wires or charged particles in motion.
• Additionally, force fields can also arise from the interaction of:
  1. Changing electric fields: Electromagnetic force fields are generated by changing electric fields, such as in the case of light or other electromagnetic waves.

  2. Spin and angular momentum: Force fields can also arise from the interaction of spin and angular momentum, such as in the case of magnetic moments or spin-orbit coupling.

• These force fields can interact with other objects and charges, resulting in forces that can cause changes in motion, energy, and momentum.

4. Similarities and differences between gravitational and electrostatic forces:

⇒ Similarities between gravitational and electrostatic forces:

1. Inverse Square Law: Both forces obey the inverse square law, meaning the force decreases with the square of the distance between objects.
2. Conservative Forces: Both forces are conservative, meaning the energy spent moving an object from one point to another is independent of the path taken.
3. Field Representation: Both forces can be represented as fields (gravitational and electrostatic fields) that surround objects.
4. Vector Nature: Both forces are vector quantities, having both magnitude and direction.
5. Superposition: Both forces follow the principle of superposition, meaning the total force is the sum of individual forces.
6. Action-at-a-Distance: Both forces appear to act over a distance, without physical contact.

⇒ Differences between gravitational and electrostatic forces:

Gravitational and electrostatic forces are two fundamental forces in physics, but they have distinct differences:

1.  Nature:
   – Gravitational force: attractive, always pulls objects towards each other
   – Electrostatic force: can be attractive (opposite charges) or repulsive (like charges)
2. Range:
   – Gravitational force: long-range, decreases with distance (1/r²)
   – Electrostatic force: short-range, decreases with distance (1/r²), but shields by other charges
3. Strength:
   – Gravitational force: relatively weak, but dominates at large scales (e.g., planets, galaxies)
   – Electrostatic force: relatively strong, dominates at small scales (e.g., atoms, molecules)
4. Mass vs Charge:
   – Gravitational force: depends on mass and distance
   – Electrostatic force: depends on charge and distance
5. Shielding:
   – Gravitational force: no shielding, always affects other masses
   – Electrostatic force: shielding by other charges, can cancel or reduce the force
6. Field Lines:
   – Gravitational force: field lines radiate from masses, never closed loops
   – Electrostatic force: field lines radiate from charges, can form closed loops
7. Particle Interaction:
   – Gravitational force: affects all particles with mass or energy
   – Electrostatic force: affects charged particles (e.g., electrons, protons)

Table 1 Some similarities and differences between gravitation and electricity

5. Coulomb’s law:

• Coulomb’s Law describes the electrostatic force between two-point charges. It states that the magnitude of the electrostatic force (F) between two-point charges (and) is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance (r) between them.
• Mathematically, Coulomb’s Law is expressed as:
• [math] F = k \frac{q_1 q_2}{r^2} [/math]
• where:
  – F is the electrostatic force
  – k is Coulomb’s constant [math] (\text{approximately}\, 8.99 \times 10^9 \, \text{Nm}^2\text{C}^{-2}) \,  k = \frac{1}{4 \pi \varepsilon_0}  [/math]
  –  and  are the magnitudes of the two-point charges
  – r is the distance between the centers of the two charges
• Coulomb’s Law applies to:
  – Stationary point charges
  – Like and unlike charges
  – Charges in vacuum or air
• This law has many applications, such as:
  – Calculating electrostatic forces between charged particles
  – Determining electric fields and potentials
  – Understanding electrostatic attraction and repulsion
• Examples:
  Calculate the force of attraction between two-point charges  and  separated by a distance of 12.2m. The charge at  is +2Cand the charge at is -1C
  Solution:
• [math] F = k\frac{ q_1 q_2}{r^2} \\
  F = \frac{1}{4\pi \varepsilon_0} \times \frac{q_1 q_2}{r^2} \\
  F = \frac{1}{4(3.14)(8.85 \times 10^{-12})} \times \frac{(2 \times 10^{-6})(-1 \times 10^{-6})}{(12.2)^2} \\
  F = \frac{1}{111.2 \times 10^{-12}} \times \frac{-2 \times 10^{-12}}{148.84} \\
  F = \frac{-2 \times 10^{-12}}{1.7 \times 10^{-8}} \\
  F = -1.2 \times 10^{-4} \, \text{N}
  [/math]
• The significance of the minus sign is to remind us that the force is attractive, but it is not really necessary to include it.

• ⇒ Potential difference:

• The work done, against an electric held, in moving unit charge from one point to a second point at a higher potential, if a charge moves from a point of higher potential to a lower potential, work is done by the electric field.
• Electric potential difference is the difference in electrical potential between two points. When the potential difference is ΔV the work alone in moving a charge Q between the two points is
• [math] \Delta W = Q \Delta V [/math]

• ⇒ Equipotential surface:

• An equipotential surface is a surface where the potential energy is constant. In other words, it’s a surface where the electric potential (voltage) or gravitational potential is the same at every point.
• Here are some key points about equipotential surfaces:
  1. Constant potential: The potential energy is the same at every point on the surface.
  2. No work done: No work is done when moving an object from one point to another on an equipotential surface.
  3. Perpendicular to field lines: Equipotential surfaces are always perpendicular to the electric or gravitational field lines.
  4. No force component: There is no force component parallel to the surface, only perpendicular to it.
  5. Surfaces can be closed or open: Equipotential surfaces can be closed (like a sphere) or open (like a plane).
  6. Used in various applications: Equipotential surfaces are used in electrostatics, gravity, and other areas of physics to analyze and visualize potential distributions.
• The equipotential surface may be linked to the electric field strength through the equation.
• [math] E = – \frac{|DeltaV} {\Deltar} [/math]
• Figure 8 equipotential surfaces are close together
• Figure 8 shows that, near the surface of the sphere, the equipotential surfaces are closer together.
• This means that both the potential gradient and the electric field strength are higher near the sphere’s surface than they are further away.
• The red lines in the diagram represent electric field lines, which point radially away from the positive sphere.
• These lines get further apart with distance from the sphere, which also shows a field diminishing with distance away from the sphere’s center.
• The field lines are always at right angles to the equipotential surfaces.
• So, when a charged particle moves along an equipotential surface, no work is done by the electric field. This can be explained using two ideas.
• First, work done is defined by
• [math] \Delta W = Q \Delta V
• or, in words, the work done on a charge is the change in potential multiplied by the charge. When ΔV = 0 (moving along an equipotential surface) the work done is zero.
• Some examples of equipotential surfaces include:
  – The surface of a sphere (in a gravitational or electrostatic field)
  – A plane (in a uniform electric or gravitational field)
  – A cylindrical surface (in a cylindrical coordinate system)

• ⇒ Similarities:

• Both gravitational and electrostatic forces have many similarities due to their inverse-square law nature. Some of the similarities include:
  1. Inverse-square law: Both forces decrease with the square of the distance between objects.
  2. Field lines: Both forces can be represented using field lines, which emerge from positive charges (or masses) and enter negative charges (or masses).
  3. Potential concept: Both forces have a potential energy associated with them, which describes the energy an object has due to its position in the field.
  4. Equipotential surfaces: Both forces have equipotential surfaces, which are surfaces where the potential energy is constant.
  5. Superposition: Both forces follow the principle of superposition, meaning the total force is the sum of individual forces.
  6. Vector nature: Both forces are vector quantities, having both magnitude and direction.
  7. Conservative forces: Both forces are conservative, meaning the energy spent moving an object from one point to another is independent of the path taken.
• These similarities highlight the deep connection between gravitational and electrostatic forces, and demonstrate how they share many common characteristics despite being distinct fundamental forces.

6. Differences: masses always attraction and charges may attract or repel:

• the differences between gravitational and electrostatic forces:
• Masses always attract:
  – Gravitational forces are always attractive, meaning that two masses will always pull each other towards each other.
  – This is because mass is always positive, and the gravitational force is proportional to the product of the two masses.
  – Gravitational attraction is responsible for:

  – Planetary orbits
  – Galaxy formation
  – Tides
  – Gravitational waves

• Charges may attract or repel:
  – Electrostatic forces can be either attractive or repulsive, depending on the nature of the charges involved.
  – Like charges (positive-positive or negative-negative) repel each other.
  – Opposite charges (positive-negative or negative-positive) attract each other.
  – Electrostatic forces are responsible for:- Attraction between opposite charges (e.g., proton-electron attraction)

  – Repulsion between like charges (e.g., electron-electron repulsion)
  – Chemical bonding (e.g., ionic, covalent, and hydrogen bonds)
  – Electric fields and potentials 

  – Gravitational forces are always attractive, while electrostatic forces can be both attractive and repulsive.
  – Mass is always positive, while charge can be positive or negative.
  – Gravitational forces are weaker than electrostatic forces at small scales (e.g., atomic and molecular levels).