DP IB Physics: SL

D. Fields

D.2 Electric and magnetic fields

DP IB Physics: SL

D. Fields

D.2 Electric and magnetic fields

Understandings
Standard level and higher level: 8 hours
a) The direction of forces between the two types of electric charge
b) Coulomb’s law as given by [math]F = k \frac{q_1 q_2}{r^2}[/math] for charged bodies treated as point charges where [math]k = \frac{1}{4 \pi \varepsilon_0}[/math]
c) The conservation of electric charge
d) Millikan’s experiment as evidence for quantization of electric charge
e) That the electric charge can be transferred between bodies using friction, electrostatic induction and by contact, including the role of grounding (earthing)
f) The electric field strength as given by [math]E = \frac{F}{q}[/math]
g) Electric field lines
h) The relationship between field line density and field strength
i) The uniform electric field strength between parallel plates as given by

[math]E = \frac{V}{d}[/math]
j) Magnetic field lines

Additional higher level: 6 hours

Understandings
Students should understand:
a) The electric potential energy [math]E_p[/math] in terms of work done to assemble the system from infinite separation
b) The electric potential energy for a system of two charged bodies as given by

[math]E_p = k \frac{q_1 q_2}{r}[/math]
c) That the electric potential is a scalar quantity with zero defined at infinity
d) That the electric potential [math]V_e[/math] at a point is the work done per unit charge to bring a test charge from infinity to that point as given by

[math]V_e = k \frac{Q}{r}[/math]
e) The electric field strength E as the electric potential gradient as given by

[math]E = \frac{\Delta V_0}{\Delta r}[/math]
f) the work done in moving a charge q in an electric field as given by

[math]W = q∆V_0[/math]
g) equipotential surfaces for electric fields
h) the relationship between equipotential surfaces and electric field lines.

 

  • a) The Direction of Forces Between Two Types of Electric Charge

  • ⇒  Types of Electric Charges:
  • – Positive charge (+)
  • – Negative charge (–)
  • Figure 1 types of electric charges
  • ⇒  Basic Rule:
  • Like charges repel, unlike charges attract.
Charges Direction of Force
+ and + Repel each other
– and – Repel each other
+ and – Attract each other
  • ⇒  Nature of Force:
  • The force is mutual: If charge A exerts a force on charge B, charge B exerts an equal and opposite force on charge A (Newton’s Third Law).
  • The force acts along the line joining the two charges.

  • b) Coulomb’s Law

  • Coulomb’s Law describes the electrostatic force between two-point charges.
  • [math]F = k \frac{q_1 q_2}{r^2}[/math]
  • Where:
  • – F: Electrostatic force (in Newtons)
  • – [math]q_1, q_2[/math]: Magnitudes of the two-point charges (in Coulombs)
  • – r: Distance between the centers of the two charges (in meters)
  • – k: Coulomb’s constant
  • [math]k = \frac{1}{4 \pi \varepsilon_0} \\
    k \approx 8.99 \times 10^9 \ \text{Nm}^2/\text{C}^2[/math]
  • – [math]ε_0[/math]: Permittivity of free space
  • [math]ε_0 = 8.85 × 10^{-12} C^2/Nm^2[/math]
  • ⇒  Interpretation:
  • – Inverse-square law: As distance increases, the force decreases with
  • – Product of charges: Larger charges exert stronger forces.
  • – Sign of charges determines whether the force is attractive or repulsive:
  • If ​[math]q_1[/math] and [math]q_2[/math] have opposite signs → force is attractive
  • If they have same signs → force is repulsive
  • ⇒  Example:
  • Two charges:
  • [math]q_1 = +2 × 10^{-6}C[/math]
  • [math]q_2 = -3 × 10^{-6}C[/math]
  • Distance r = 0.05m
  • [math]F = (8.99 \times 10^9) \frac{(2 \times 10^{-6})(3 \times 10^{-6})}{(0.05)^2} \\
    F = 21.6 \text{ N (attractive force)}[/math]

  • c) Conservation of Electric Charge

  • Principle:
  • Electric charge can neither be created nor destroyed, only transferred from one body to another.
  • This is a universal law of physics, valid in all processes including chemical, physical, and nuclear reactions.
  • Implications:
  • – The net charge in an isolated system is constant.
  • – If a neutral object gains a negative charge, an equal positive charge must be left behind somewhere else.
  • – Even in pair production or annihilation in particle physics, the total charge remains unchanged.
  • Examples:
  • 1. Rubbing a balloon on hair:
  • – Electrons are transferred from hair to balloon.
  • – Balloon becomes negatively charged; hair becomes positively charged.
  • – Total charge remains zero.
  • Figure 2 Rubbing a balloon on hair
  • 2. Charging by induction:
  • – A charged object induces redistribution of charges in another object.
  • – Charges are not created—only moved around.
  • 3. Nuclear reaction (e.g., beta decay):
  • – A neutron becomes a proton and emits an electron.
Concept Description Equation
Force between charges Like charges repel, unlike attract Direction depends on charge signs
Coulomb’s Law Electrostatic force between two-point charges [math]F = k \frac{q_1 q_2}{r^2}[/math]
Constant k Coulomb’s constant [math]8.99 × 10^9 Nm^2/C^2[/math]
Conservation of Charge Total charge remains constant in any process ΔQ = 0 in isolated systems
  • ⇒  Real-World Applications:
  • – Capacitors: Store charge using attraction between opposite charges.
  • – Electrostatics in printers: Uses attractive and repulsive forces to apply toner.
  • – Particle accelerators: Rely on Coulomb forces to control charged particles.
  • – Lightning: Massive charge separation and redistribution during a storm.

  • d) Millikan’s Oil Drop Experiment and Quantization of Electric Charge

  • ⇒  Purpose:
  • Millikan’s experiment demonstrated that electric charge is quantized—that is, charge always occurs in integer multiples of a fundamental unit, the elementary charge e.
  • Figure 3 Millikan’s Oil drop
  • ⇒  Setup:
  • – Tiny oil droplets were sprayed into a chamber.
  • – Some droplets became charged due to friction or ionization.
  • – Droplets entered a region between two horizontal metal plates.
  • – By adjusting the voltage across the plates, Millikan was able to balance the gravitational force pulling the droplet down with the electric force pushing it up.
  • ⇒  Forces Involved:
  • At equilibrium:
  • [math]F_\text{electric} = F_\text{gravity} \\
    qE = mg[/math]
  • Where:
  • – q is the charge on the oil drop
  • – E is the electric field between the plates
  • – m is the mass of the oil drop
  • – g is acceleration due to gravity
  • From the known mass, gravity, and electric field, Millikan calculated q for many oil drops and found:
  • The charge on each droplet was always a multiple of [math]1.6 × 10^{-19} C[/math]
  • ⇒ Conclusion:
  • Electric charge is quantized. The smallest possible charge is [math]e = 1.6 × 10^{-19} C[/math] (the charge of a single electron or proton).

  • e) Electric Charge Transfer

  • Electric charge can be transferred between objects in three main ways:
  • 1. By Friction (Charging by Rubbing):
  • – Electrons are transferred from one object to another through rubbing.
  • Example:
  • Rubbing a balloon on hair → hair becomes positively charged, balloon becomes negatively charged.
  • Triboelectric effect: some materials are more likely to lose or gain electrons when rubbed.
  • 2. By Contact (Conduction):
  • – When a charged object touches a neutral object, electrons move to equalize the charge.
  • – The neutral object becomes charged with the same sign as the original object.
  • – Works best if the second object is a conductor.
  • Figure 4 Conduction
  • 3. By Induction (Without Touching):
  • – A charged object brought near a conductor causes electrons to redistribute within it.
  • – The conductor becomes polarized (positive on one end, negative on the other).
  • – If one side is grounded (connected to Earth), charge can leave or enter, permanently charging the object.
  • ⇒  Role of Grounding (Earthing):
  • The Earth acts as a reservoir of electrons.
  • Grounding allows excess charge to flow to or from the Earth.
  • – Grounding a positive object → electrons flow into it from Earth.
  • – Grounding a negative object → electrons flow out into Earth.
  • Used to neutralize objects or to safely discharge build-up of static charge (important in electronics and lightning protection).
  • Figure 5 Role of grounding

  • f) Electric Field Strength

  • The electric field strength E at a point is the force per unit charge experienced by a small positive test charge placed at that point.
  • [math]E = \frac{F}{q}[/math]
  • Where:
  • E = Electric field strength (in N/C)
  • F = Force on the test charge (in N)
  • q = Magnitude of the test charge (in C)
  • Figure 6 Electric field strength
  • The direction of the electric field is the direction of the force on a positive charge.
  • If F is known, and you know the charge q, you can determine how strong the field is at that point.
  • ⇒   Uniform Electric Field:
  • Between two parallel plates of opposite charges:
  • [math]E = \frac{V}{d}[/math]
  • Where:
  • – V = Potential difference between plates
  • – d = Distance between plates
  • This gives a constant (uniform) electric field.
  • Figure 7 Uniform electric field
  • ⇒  Point Charges:
  • The electric field created by a single point charge is given by:
  • [math]E = k \frac{q}{r^2} [/math]
  • Where:
  • – [math]k = 8.99 \times 10^9 \frac{\text{Nm}^2}{\text{C}^2}[/math]
  • – q = Source charge
  • – r = Distance from the charge

  • g) Electric Field Lines

  • Electric field lines are visual representations that show the direction and strength of the electric field in space.
  • ⇒  Properties of Electric Field Lines:
    1. Direction: They point away from positive charges and toward negative charges.
    2. Lines never cross: If they did, it would mean the field has two directions at one point, which is not possible.
    3. Field line density: The closer the lines, the stronger the electric field in that region.
    4. Field lines begin or end on charges:
  • – Begin at +ve charges
  • -End at –ve charges
    1. Perpendicular to surfaces: Field lines are always perpendicular to the surface of conductors.
  • Figure 8 Electric field line
  • ⇒   Visualization Examples:
Charge Setup Field Line Pattern
Single +ve charge Lines radiate outward symmetrically
Single –ve charge Lines radiate inward symmetrically
Dipole (+ and –) Curved lines go from + to –, forming loops

  • h) Relationship Between Field Line Density and Field Strength

  • The density of electric field lines in a diagram is proportional to the field strength.
  • – More lines per area = Stronger field
  • – Fewer lines per area = Weaker field
  • This allows visual estimates of field strength without calculation.

  •   i) Uniform Electric Field Between Parallel Plates

  • When two large, flat, oppositely charged plates are placed parallel to each other, they create a uniform electric field between them.
  • [math]E = \frac{V}{d}[/math]
  • Where:
  • – E: Electric field strength (in N/C)
  • – V: Potential difference between the plates (in volts)
  • – d: Distance between the plates (in meters)
  • Figure 9 Uniform electric field between parallel plates
  • ⇒   Characteristics:
  • Field lines are parallel and equally spaced between the plates.
  • Field is uniform in strength and direction.
  • Near the edges, the field becomes non-uniform (fringing effect).

  • j) Magnetic Field Lines

  • Magnetic field lines represent the direction and strength of the magnetic field in space.
  • They are used to visualize magnetic forces, especially around magnets and current-carrying wires.
  • Figure 10 Magnetic field lines
  • ⇒  Properties of Magnetic Field Lines:
  • 1. Form closed loops: They go from the north pole to the south pole outside the magnet, and south to north inside the magnet.
  • 2. Field line direction: The direction a north pole would move in the field.
  • 3. Never intersect: Just like electric field lines.
  • 4. Closer lines = stronger field: Density again indicates field strength.
  • ⇒  Magnetic Field Sources:
  • Bar magnets: Field lines loop from N to S outside the magnet.
  • Current-carrying wires:
  • – Straight wire: Circular magnetic field around the wire.
  • – Solenoid (coil): Field inside is uniform and similar to a bar magnet.

  • Additional higher level: 6 hours

  • a)   Electric Potential Energy Ep:

  • Electric potential energy Ep​ is the energy stored in a system of electric charges due to their relative positions.
  • It is also defined as the work done to assemble the system from infinite separation to a given configuration.
  • When we bring two charges closer together from a very far distance (infinity), we either do work on the system or the system does work on us, depending on whether the charges repel or attract.
  • ⇒  Work Done from Infinity:
  • By convention, we say that:
  • – The electric potential energy between two charges at infinite separation is zero.
  • – Any work done to bring them closer adds to or removes energy from the system.
  • Figure 11 Electric charge work done form infinity

  • b)   Electric Potential Energy Between Two Point Charges

  • [math]E_p = k \frac{q_1 q_2}{r}[/math]
  • Where:
  • – ​[math]E_p[/math] = Electric potential energy (in joules)
  • – [math]k = \frac{1}{4 \pi \varepsilon_0} \approx 8.99 \times 10^9 \ \frac{\text{Nm}^2}{\text{C}^2}[/math]
  • – ​[math]q_1, q_2[/math] = Magnitudes of the two-point charges (in coulombs)
  • – r = Distance between the charges (in meters)
  • ⇒   Sign of Ep ​:
Type of Charges Sign of Ep ​Meaning
Both Positive or Both Negative Positive Work must be done to bring them closer (they repel)
One Positive, One Negative Negative They attract, so the system releases energy as they come closer
  • This is consistent with energy conservation: potential energy increases when you store energy in the system (by pushing like charges together), and decreases when energy is released (by allowing unlike charges to attract).

  • c)    Electric Potential V: Scalar and Zero at Infinity

  • o Electric potential V at a point in space is the electric potential energy per unit charge at that point.
  • [math]V = \frac{E_p}{q} \\
    V = k \frac{Q}{r}[/math]
  • Where:
  • – V = Electric potential (in volts = J/C)
  • – Q = Source charge creating the potential
  • – q = Test charge
  • – r = Distance from the source charge
  • ⇒ Scalar Quantity:
  • Electric potential is a scalar: it only has magnitude, not direction.
  • Unlike the electric field (a vector), you just add potentials algebraically (positive and negative values), no vector sum needed.
  • ⇒   Zero Defined at Infinity:
  • – The potential at infinity is defined to be zero.
  • – As you move a charge closer to another, the potential changes depending on the sign of the charge.
  • ⇒   Relationship Between Ep and V:
  • [math]E_p = qV[/math]
  • This connects the two concepts: if you know the potential V at a point, the energy a charge q would have at that point is simply [math]qV[/math]
  • ⇒   Real-World Analogy
  • Think of lifting a book onto a shelf:
  • The floor is like infinity (zero potential energy).
  • The shelf is closer to a source of force (like a charge).
  • Lifting the book is like bringing a charge closer to another—it stores energy.

  • d)   Electric Potential [math]V_e[/math]

  • The electric potential Ve at a point is the work done per unit charge to bring a small positive test charge from infinity to that point in an electric field.
  • [math]V_e = \frac{W}{q} \\ V_e = k \frac{Q}{r}[/math]
  • Where:
  • – [math]V_e[/math]​: Electric potential at a point (in volts, V = J/C)
  • – W: Work done in bringing the charge from infinity
  • – q: Small test charge (in coulombs, C)
  • – Q: Source charge creating the field
  • – r: Distance from the source charge to the point
  • – k: Coulomb’s constant, [math]k = \frac{1}{4 \pi \epsilon_0} \approx 8.99 \times 10^9 \text{ N·m}^2/\text{C}^2[/math]
  • Scalar quantity — it has magnitude but no direction.
  • Defined relative to infinity, where [math]V_e = 0[/math].
  • Depends only on the source charge and distance, not on the test charge.
  • ⇒  Sign Convention:
Situation Ve ​ Sign
Near a positive charge Positive
Near a negative charge Negative

  • e)    Electric Field Strength as Potential Gradient

  • The electric field strength at a point is equal to the rate of change of electric potential with respect to distance — also called the potential gradient.
  • [math]E = -\frac{\Delta V}{\Delta r}[/math]
  • Where:
  • – E: Electric field strength (in N/C or V/m)
  • – ΔV: Change in electric potential (volts)
  • – Δr: Distance over which the potential changes (meters)
  • The negative sign indicates that the electric field points in the direction of decreasing potential — i.e., from high to low potential.
  • In a Uniform Field Between Parallel Plates:
  • If the field is uniform, as between two parallel plates:
  • [math]E = \frac{V}{d}[/math]
  • Where:
  • – V: Potential difference between the plates
  • – d: Distance between the plates

  • f)     Work Done in Moving a Charge in an Electric Field

  • The work done in moving a charge q across a potential difference ΔV in an electric field is given by:
  • [math]W = qΔV[/math]
  • Where:
  • – W: Work done (in joules, J)
  • – q: Charge being moved (in coulombs, C)
  • – ΔV: Change in electric potential (in volts)
  • Figure 12 direction of electric charges
  • ⇒  Interpretation:
  • If ΔV is positive and the charge is positive, the field does work on the charge (it gains energy).
  • If ΔV is negative, the charge is moving against the field — you must do work on it to move it.
  • This equation is fundamental in understanding energy changes in capacitors, batteries, and circuits.

  • (g) Equipotential Surfaces for Electric Fields

  • Equipotential surfaces are imaginary surfaces in an electric field where the electric potential is the same at every point on the surface.
  • That means no work is required to move a charge along this surface because the potential energy doesn’t change.
  • Figure 13 Equi-Potential Surfaces for electric field
  • ⇒  Properties of Equipotential Surfaces:
Feature Description
Constant Potential Every point on the surface has the same electric potential
No Work Done Moving a charge along the surface requires no energy

[math]W = q∆V = 0[/math]
Perpendicular to Field Lines They are always at right angles to electric field lines
Closer Spacing = Stronger Field Where equi-potentials are closer together, the electric field is stronger
3D Shapes They can be spherical, planar, cylindrical, etc., depending on the source of the field
  • ⇒  Examples of Equipotential Surfaces:
  • 1. Point Charge:
  • – The equipotential surfaces are concentric spheres centered around the charge.
  • – For a positive point charge, the potential decreases as you move outward.
  • 2. Uniform Field (e.g. between parallel plates):
  • – The equi-potentials are parallel planes, perpendicular to the electric field.
  • – Each plane has a different potential value.

  • h)   Relationship Between Equipotential Surfaces and Electric Field Lines

  • ⇒   Electric Field Lines:
  • Electric field lines show the direction of force on a positive test charge. They point from positive to negative.
  • ⇒   Relationship Between Field Lines and Equipotential Surfaces:
Aspect Relationship
Perpendicularity Electric field lines are always perpendicular (90°) to equipotential surfaces.
No Component Along Surface There is no electric field along an equipotential surface — only across it.
Work Done No work is done moving along an equipotential

(since [math]\vec{E} \cdot \vec{d} = 0[/math])
Gradient and Field Strength The steeper the potential change (closer equi-potentials), the stronger the electric field
  • ⇒   Link to Field Strength:
  • [math]E = -\frac{\Delta V}{\Delta r}[/math]
  • – The electric field strength is the rate of change of potential across equi-potentials.
  • – So, where equi-potentials are closer, the potential changes more quickly with distance, meaning a stronger electric field.
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