Welcome to Force Fields!
Welcome to one of the most fascinating topics in A2 Physics! Have you ever wondered how the Moon stays in orbit around the Earth without any physical cables connecting them? Or why your hair stands on end when you pull a woolly jumper over your head? The answer lies in force fields.
In this chapter, we will demystify how objects exert forces on each other from a distance. We will look closely at gravitational fields and electric fields, explore the mathematics that describes them, and discover just how remarkably similar they are. Don't worry if equations with symbols like \(\varepsilon_0\) or \(G\) look intimidating at first — we will break down every single concept step-by-step!
1. What is a Force Field?
A force field is defined as a region of space in which an object experiences a non-contact force.
Whenever an object with a specific property enters a field, it feels a push or a pull:
• An object with mass experiences a force in a gravitational field.
• An object with electric charge experiences a force in an electric field.
• A moving charge or magnetic material experiences a force in a magnetic field.
Representing Fields: Field Lines
We visualize invisible force fields using field lines (also called lines of force). Field lines follow simple rules:
1. The direction of the arrow shows the direction of the force exerted on a test object.
2. The density (closeness) of the lines shows the field strength. Closer lines mean a stronger field; lines spreading further apart mean a weaker field.
3. Field lines never cross each other.
Uniform vs Radial Fields
There are two main field shapes you need to know for your exam:
1. Radial Fields: The field lines spread outwards or inwards like the spokes of a bicycle wheel. The strength of the field decreases as you move further away from the central object (an inverse-square relationship).
Example: The gravitational field around planet Earth, or the electric field surrounding an isolated point charge.
2. Uniform Fields: The field lines are parallel, straight, and equally spaced. This means the field has the exact same strength and direction at every point.
Example: The electric field between two parallel oppositely charged metal plates, or the gravitational field very close to the surface of the Earth across a small area.
Key Takeaway: Field lines tell you two things at a glance: direction (arrows) and strength (closeness of lines).
2. Gravitational Fields
Newton's Law of Universal Gravitation
Sir Isaac Newton realized that every point mass attracts every other point mass in the universe with a gravitational force. The magnitude of this force depends on the masses of the two objects and the distance between them.
Newton's Law states: The attractive gravitational force \(F\) between two point masses \(M\) and \(m\) is directly proportional to the product of their masses and inversely proportional to the square of the distance \(r\) between their centres.
\(F = \frac{G M m}{r^2}\)
Where:
• \(F\) = Gravitational force (measured in Newtons, \(\text{N}\))
• \(G\) = Universal Gravitational Constant \(= 6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}\)
• \(M\) and \(m\) = The two interacting masses (measured in \(\text{kg}\))
• \(r\) = Distance between the centres of the two masses (measured in \(\text{m}\))
Memory Tip: Always measure \(r\) from the centre of mass to the centre of mass, not surface to surface!
Gravitational Field Strength (\(g\))
Definition: The gravitational field strength \(g\) at a point in a field is defined as the gravitational force per unit mass acting on a small test mass placed at that point.
\(g = \frac{F}{m}\)
Units: \(\text{N kg}^{-1}\) (which is equivalent to \(\text{m s}^{-2}\)).
By substituting Newton's law (\(F = \frac{G M m}{r^2}\)) into this definition, we find the formula for the gravitational field strength created by a mass \(M\):
\(g = \frac{G M}{r^2}\)
Important Note: Notice that \(g\) is inversely proportional to \(r^2\) (\(g \propto \frac{1}{r^2}\)). If you double the distance from the centre of a planet (\(2r\)), the field strength becomes \(\frac{1}{2^2} = \frac{1}{4}\) of its original value!
Gravitational Potential (\(V_g\))
Definition: The gravitational potential \(V_g\) at a point is the work done per unit mass in bringing a small test mass from infinity to that point.
\(V_g = -\frac{G M}{r}\)
Units: \(\text{J kg}^{-1}\).
Why is Gravitational Potential always negative?
• At an infinite distance (\(r = \infty\)), the gravitational force is zero, so potential is defined as zero at infinity (\(V_g = 0\)).
• Gravity is an attractive force. As an object is brought closer to a mass from infinity, energy is released (the field does work on the mass). Therefore, its potential energy drops below zero, making it negative.
Relationship between Field Strength and Potential:
The gravitational field strength is the negative potential gradient:
\(g = -\frac{\Delta V_g}{\Delta r}\)
Key Takeaway: Gravitational force is always attractive. Both force and field strength obey an inverse-square law with distance.
3. Electric Fields
Coulomb's Law
Just as masses exert gravitational forces on each other, electric charges exert electrostatic forces. Charles-Augustin de Coulomb discovered the law governing these forces.
Coulomb's Law states: The electrostatic force \(F\) between two point charges \(Q_1\) and \(Q_2\) is directly proportional to the product of their charges and inversely proportional to the square of the distance \(r\) between their centres.
\(F = \frac{Q_1 Q_2}{4 \pi \varepsilon_0 r^2}\)
Where:
• \(F\) = Electrostatic force (in \(\text{N}\))
• \(Q_1, Q_2\) = Charges (in Coulombs, \(\text{C}\))
• \(r\) = Distance between the centres of the charges (in \(\text{m}\))
• \(\varepsilon_0\) = Permittivity of free space \(= 8.85 \times 10^{-12} \text{ F m}^{-1}\)
• The constant factor \(\frac{1}{4 \pi \varepsilon_0} \approx 8.99 \times 10^9 \text{ N m}^2 \text{ C}^{-2}\)
Did you know? Unlike gravity (which is only attractive), electrostatic forces can be attractive (between opposite charges: \(+-\)) or repulsive (between like charges: \(++\) or \(--\)).
Electric Field Strength (\(E\))
Definition: The electric field strength \(E\) at a point is the force per unit positive charge acting on a stationary test charge placed at that point.
\(E = \frac{F}{q}\)
Units: \(\text{N C}^{-1}\) or \(\text{V m}^{-1}\).
Radial Electric Field
For a point charge \(Q\), the electric field strength at distance \(r\) is:
\(E = \frac{Q}{4 \pi \varepsilon_0 r^2}\)
Uniform Electric Field (Parallel Plates)
When two parallel metal plates are separated by a distance \(d\) and connected to a potential difference \(V\), a uniform electric field is created between them:
\(E = \frac{V}{d}\)
The force acting on a charge \(q\) placed inside this uniform field is simply:
\(F = q E = \frac{q V}{d}\)
Electric Potential (\(V\))
Definition: The electric potential \(V\) at a point is the work done per unit positive charge in bringing a small positive test charge from infinity to that point.
\(V = \frac{Q}{4 \pi \varepsilon_0 r}\)
Units: Volts (\(\text{V}\)) or Joules per Coulomb (\(\text{J C}^{-1}\)).
• Near an isolated positive charge, \(V\) is positive (you must do work to push a positive test charge closer).
• Near an isolated negative charge, \(V\) is negative (the negative charge attracts the positive test charge).
Key Takeaway: Electric field strength is the force per unit positive charge. In a uniform field, \(E = \frac{V}{d}\); in a radial field, \(E = \frac{Q}{4 \pi \varepsilon_0 r^2}\).
4. Comparing Gravitational and Electric Fields
Examiners love asking you to compare these two fields! Here is a simple comparison breakdown:
Similarities:
• Both are non-contact forces that can act through a vacuum.
• Both follow an inverse-square law for radial fields (\(F \propto \frac{1}{r^2}\)).
• Both have field strengths defined as force per unit property (\(g = \frac{F}{m}\) and \(E = \frac{F}{q}\)).
• Both have potential defined as work done per unit property from infinity (\(V = \frac{W}{m}\) and \(V = \frac{W}{q}\)).
• Both can be represented by field lines and equipotential surfaces.
Differences:
• Nature of force: Gravitational forces are always attractive. Electric forces can be either attractive or repulsive.
• Property causing the field: Gravitational fields are caused by mass; electric fields are caused by charge.
• Shielding: Electric fields can be shielded or blocked (e.g., using a Faraday cage); gravitational fields cannot be shielded.
• Relative Strength: Electrostatic forces are immensely stronger than gravitational forces at the atomic scale (by a factor of around \(10^{36}\)!).
5. Motion of Charged Particles in Uniform Electric Fields
When a charged particle (like an electron or proton) enters a uniform electric field perpendicular to the field lines, it follows a parabolic path. This is directly analogous to projectile motion under gravity!
Step-by-Step Analysis:
Let's consider an electron of mass \(m\) and charge \(-e\) entering horizontally with speed \(v_x\) between two parallel plates:
1. Horizontal Motion: No force acts horizontally, so horizontal velocity remains constant: \(v_x = \text{constant}\). Time in the field is \(t = \frac{L}{v_x}\) (where \(L\) is the plate length).
2. Vertical Motion: A constant vertical electrostatic force acts: \(F = e E = \frac{e V}{d}\).
3. Vertical Acceleration: Using Newton's second law (\(F = m a\)):
\(a = \frac{F}{m} = \frac{e E}{m} = \frac{e V}{m d}\)
4. Vertical Deflection (\(y\)): Using \(s = u t + \frac{1}{2} a t^2\) (with initial vertical velocity \(u_y = 0\)):
\(y = \frac{1}{2} a t^2 = \frac{1}{2} \left(\frac{e E}{m}\right) \left(\frac{L}{v_x}\right)^2\)
Common Exam Question: Why do electrons deflect more than protons with the same initial speed?
Even though both experience the same magnitude of electric force (\(F = e E\)), the electron has a vastly smaller mass (\(m_e \ll m_p\)), so its acceleration (\(a = \frac{F}{m}\)) and deflection are much greater!
6. Common Mistakes to Avoid
• Confusing \(r\) and \(r^2\): Force and Field Strength have \(r^2\) on the denominator; Potential has \(r\) on the denominator. Double-check your formula sheet before calculating!
• Distance unit conversion: Distances are often given in centimetres (\(\text{cm}\)) or kilometres (\(\text{km}\)). Always convert to metres (\(\text{m}\)) before squaring: \(1 \text{ km} = 10^3 \text{ m}\), \(1 \text{ cm} = 10^{-2} \text{ m}\).
• Forgetting the negative sign for Gravitational Potential: Gravitational potential is always negative: \(V_g = -\frac{G M}{r}\).
• Direction of arrows on electric field lines: Remember that arrows show the direction of force on a positive charge. Lines point away from positive charges and towards negative charges.
7. Chapter Summary Checklist
Before you move on to past paper questions, ensure you can:
• Define gravitational field strength (\(g\)) and electric field strength (\(E\)).
• State and apply Newton's Law of Universal Gravitation (\(F = \frac{G M m}{r^2}\)) and Coulomb's Law (\(F = \frac{Q_1 Q_2}{4 \pi \varepsilon_0 r^2}\)).
• Calculate the field strength for both radial and uniform fields.
• Explain the concepts of gravitational and electric potentials.
• Describe and calculate the parabolic trajectory of a charged particle passing through a uniform electric field.
• Compare the key similarities and differences between gravitational and electric fields.