Welcome to Fields: Understanding Forces Without Touching!
Welcome to the exciting world of Fields! In this chapter, we step away from contact forces like friction and tension and dive into how objects can influence each other over vast distances—this is often called "action at a distance."
We will explore three fundamental fields: gravitational, electric, and magnetic. Understanding fields is crucial because they are the mechanism by which forces like gravity and electromagnetism operate, explaining everything from why planets orbit the Sun to how an electric motor works.
What is a Field?
Think of a field as a region of space where a test object experiences a force.
- Field Source: The object creating the disturbance (e.g., a massive star creates a gravitational field).
- Test Object: The object used to detect the field (e.g., a small mass or a test charge).
- Field Lines: We use field lines to visualize the field. They show the direction the test object would move if placed there, and the density of the lines indicates the strength of the field.
Section D.1: Gravitational Fields (SL & HL)
The gravitational field is created by mass and influences other masses.
Key Concept: The Inverse Square Law
Both gravity and electric forces follow the inverse square law. This means the force decreases dramatically as the distance increases. If you double the distance, the force becomes only one-quarter (\(1/2^2\)) of the original force.
Universal Law of Gravitation (HL Depth)
The gravitational force \(F_G\) between two point masses, \(m_1\) and \(m_2\), separated by a distance \(r\) is given by:
\[ F_G = G \frac{m_1 m_2}{r^2} \]
- \(G\) is the Universal Gravitational Constant (\(6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}\)).
- This is a vector force, always attractive.
Gravitational Field Strength (\(g\))
Field strength (\(g\)) is defined as the force per unit mass experienced by a small test mass \(m\) placed at that point. It is a vector quantity, measured in \(\text{N kg}^{-1}\) (which is equivalent to \(\text{m s}^{-2}\)).
\[ g = \frac{F_G}{m} \]
Analogy: If a large mass creates a dent in the fabric of space (like a bowling ball on a trampoline), the gravitational field strength \(g\) tells you how steep that dent is at any point.
Gravitational Field Strength (HL Derivation): By substituting the force equation, we find the field strength around a point mass \(M\):
\[ g = G \frac{M}{r^2} \]
HL: Gravitational Potential Energy (\(E_p\))
Gravitational Potential Energy (\(E_p\)) is the energy required to move a mass \(m\) from infinity (where potential energy is defined as zero) to a point in the field.
\[ E_p = -G \frac{M m}{r} \]
Why the negative sign? Gravity is always attractive. Work must be done against the field to separate the masses. Therefore, the potential energy is always negative, reaching a maximum of zero only at infinite separation.
HL: Gravitational Potential (\(V_g\))
Gravitational Potential (\(V_g\)) is the potential energy per unit mass. It is a scalar quantity, measured in \(\text{J kg}^{-1}\).
\[ V_g = \frac{E_p}{m} = -G \frac{M}{r} \]
Memory Aid: Potential \(V_g\) is related to field strength \(g\) just as energy \(E_p\) is related to force \(F\).
Key Takeaway (D.1): The force and field strength decrease with the square of the distance (\(1/r^2\)). Gravitational potential and potential energy (HL) are negative and increase (become less negative) as distance increases.
Section D.2: Electric and Magnetic Fields (SL & HL)
Electric Fields
Electric fields are created by electric charges and influence other charges.
Coulomb’s Law (Force between charges)
The electric force \(F_E\) between two point charges, \(q_1\) and \(q_2\), separated by a distance \(r\), is given by:
\[ F_E = k \frac{q_1 q_2}{r^2} \]
- \(k\) is the Coulomb constant (\(8.99 \times 10^9 \text{ N m}^2 \text{ C}^{-2}\)).
- This is an inverse square law, similar to gravity.
- The force is attractive if charges are opposite, and repulsive if charges are the same.
Electric Field Strength (\(E\))
Electric field strength (\(E\)) is the force per unit positive test charge (\(q\)) experienced at that point. It is a vector quantity, measured in \(\text{N C}^{-1}\) (or \(\text{V m}^{-1}\)).
\[ E = \frac{F_E}{q} \]
Electric Field Strength around a Point Charge (HL Derivation):
\[ E = k \frac{Q}{r^2} \]
Field lines originate from positive charges and terminate on negative charges.
Uniform Electric Fields (Parallel Plates)
In the space between two parallel, oppositely charged plates, the electric field is uniform (constant strength and direction).
In a uniform field, the field strength \(E\) is related to the potential difference \(V\) across the plates and their separation \(d\):
\[ E = \frac{V}{d} \]
HL: Electric Potential Energy (\(E_p\)) and Electric Potential (\(V_e\))
Similar to gravity, electric charges possess potential energy due to their position in an electric field.
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Electric Potential Energy (\(E_p\)): The work done to move a charge \(q\) from infinity to a point in the field.
\[ E_p = k \frac{Q q}{r} \]
-
Electric Potential (\(V_e\)): The potential energy per unit positive charge (\(\text{J C}^{-1}\) or \(\text{Volts}\)).
\[ V_e = \frac{E_p}{q} = k \frac{Q}{r} \]
Important difference from gravity: \(V_e\) can be positive (around a positive source charge) or negative (around a negative source charge).
Potential Gradient (HL): The electric field strength is the negative gradient of the electric potential:
\[ E = -\frac{\Delta V}{\Delta x} \]
This means charges accelerate from high potential to low potential.Magnetic Fields (\(B\))
Magnetic fields are created by moving charges (currents) and influence other moving charges.
- Magnetic field strength (Magnetic Flux Density, \(B\)) is a vector, measured in Tesla (T).
- Field lines always form closed loops (unlike electric fields, which start and end on charges).
- Field lines run from the North pole to the South pole outside a magnet.
- Sources of \(B\)-fields: Permanent magnets and current-carrying wires (electromagnets).
Quick Review (D.2): Electric forces depend on charge (attractive or repulsive) and follow the inverse square law. Magnetic fields are caused by motion and form continuous loops.
Section D.3: Motion in Electromagnetic Fields (SL & HL)
When a charged particle moves through a magnetic field, it experiences a force. This force is essential for motors and mass spectrometers.
Force on a Charged Particle
The force (\(F\)) experienced by a charge (\(q\)) moving with velocity (\(v\)) in a magnetic field (\(B\)) is given by the Lorentz force component:
\[ F = q v B \sin\theta \]
- \(\theta\) is the angle between the velocity vector (\(v\)) and the magnetic field vector (\(B\)).
- If the charge moves parallel (\(\theta=0\)) or antiparallel (\(\theta=180^\circ\)) to the field, the force is zero.
- The force is maximum when the charge moves perpendicular (\(\theta=90^\circ\)) to the field.
Determining Direction: Fleming's Left-Hand Rule
Since the magnetic force is always perpendicular to both \(v\) and \(B\), we use a vector rule to find its direction:
Use your left hand (for positive charges or conventional current):
- Thumb: Force (\(F\))
- Forefinger: Magnetic Field (\(B\)) 3. Centre Finger: Current (\(I\)) or Velocity (\(v\))
If the charge is negative (like an electron), the force direction is opposite to what the rule predicts, or you can use the center finger to point opposite to the electron's motion.
Force on a Current-Carrying Conductor
Since a current is just a flow of charges, a wire carrying a current (\(I\)) through a magnetic field (\(B\)) also experiences a force:
\[ F = B I L \sin\theta \]
- \(L\) is the length of the wire inside the field.
- This principle is the basis for all electric motors.
Application: Circular Motion in a Magnetic Field
When a charged particle moves perpendicularly to a uniform magnetic field, the magnetic force (\(F = qvB\)) acts as the centripetal force.
Setting the forces equal (\(F_{\text{magnetic}} = F_{\text{centripetal}}\)):
\[ qvB = \frac{m v^2}{r} \]
We can rearrange this to find the radius (\(r\)) of the circular path:
\[ r = \frac{m v}{q B} \]
Did you know? This relationship is used in devices called mass spectrometers to separate ions based on their mass-to-charge ratio (\(m/q\)). Lighter ions trace smaller circles.
Common Mistake to Avoid: The magnetic force never changes the speed of the particle, only its direction. Since the force is always perpendicular to the velocity, the work done by the magnetic force is zero, thus kinetic energy is constant.
Key Takeaway (D.3): Moving charges and currents experience a force in a magnetic field. This force is maximum when motion/current is perpendicular to the field and causes charged particles to move in circles.
Section D.4: Electromagnetic Induction (HL ONLY)
Attention HL students! This final section deals with how we can generate electricity using magnetic fields—the principle behind electric generators and transformers.
Electromagnetic Induction is the process of generating an electromotive force (e.m.f.) by moving a conductor through a magnetic field, or by changing the magnetic field passing through a loop of wire.
Magnetic Flux (\(\Phi\))
Magnetic Flux (\(\Phi\)) is a scalar quantity that measures the total number of magnetic field lines passing through a given area.
\[ \Phi = B A \cos\theta \]
- \(B\) is the magnetic field strength (Tesla, T).
- \(A\) is the area of the loop (\(\text{m}^2\)).
- \(\theta\) is the angle between the field lines (\(B\)) and the normal (perpendicular) to the area (\(A\)).
- The unit of magnetic flux is the Weber (\(\text{Wb}\)). (\(1 \text{ Wb} = 1 \text{ T m}^2\)).
Maximum flux occurs when the area is perpendicular to the field (\(\cos 0^\circ = 1\)). Zero flux occurs when the area is parallel to the field (\(\cos 90^\circ = 0\)).
Faraday's Law of Induction
Faraday's law states that the magnitude of the induced e.m.f. (\(\varepsilon\)) is directly proportional to the rate of change of magnetic flux linkage.
For a coil with \(N\) turns:
\[ \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \]
- This is the foundation of electrical power generation. To generate a large e.m.f., you need a large number of turns (\(N\)) and a rapid change in flux (\(\frac{\Delta \Phi}{\Delta t}\)).
Lenz's Law (The Origin of the Negative Sign)
The negative sign in Faraday's Law is a consequence of Lenz's Law.
Lenz’s Law states that the direction of the induced current (or e.m.f.) is always such as to oppose the change in magnetic flux that caused it.
Analogy: Lenz’s law is like a stubborn child. If you try to push a magnet into a coil (increasing flux), the induced current creates its own magnetic field that pushes the magnet back out. If you try to pull it out (decreasing flux), the induced current creates a field that tries to pull it back in.
Why is this necessary? If the induced current did not oppose the change, it would reinforce it, leading to a perpetual increase in energy, violating the Law of Conservation of Energy.
Key Takeaway (D.4 HL): Changing magnetic flux induces an e.m.f. (Faraday's Law). The direction of the induced e.m.f. always resists this change (Lenz's Law).