Charge and field

 Module 6: Field and particle physics

6.1 Fields
6.1.2

Charge and field

a)   Describe and explain:

I)                   Uniform electric field E = V/d

II)                The electric field of a charged object, and the force on a charge in an electric field; inverse square law for point charge

III)             Electrical potential energy and electric potential due to a point charge; 1/r relationship

IV)             Evidence for discreteness of charge on electron

V)                The force on a moving charged particle due to a uniform magnetic field

VI)             Similarities and differences between electric and gravitational fields.

b)   Make appropriate use of:

I)                   The terms: charge, electric field, electric potential, equipotential surface, electron volt

by sketching and interpreting:

II)                Graphs showing electric potential as area under a graph of electric field versus distance, graphs showing changes in electric potential energy as area under a graph of electric force versus distance between two distance values.

III)             Graphs showing force as related to the tangent of a graph of electric potential energy versus distance, graphs showing field strength as related to the tangent of a graph of electric potential versus distance

IV)             Diagrams of electric fields and the corresponding equipotential surfaces.

c)    Make calculations and estimates involving:

I)                   For radial components

[math]F_{\text{electric}} =  \frac{kqQ}{r^2}, \qquad E_{\text{electric}} = \frac{F_{\text{electric}}}{q} \\ E_{\text{electric}} = \frac{kQ}{r^2}  \left[ k = \frac{1}{4 \pi \varepsilon_0} \right] [/math]

II)                [math]E_{\text{electric}} = -\frac{dV_{\text{electric}}}{dr}, \quad E_{\text{electric}} = \frac{V}{d} \quad \text{(for a uniform field)}[/math]

III)              [math]\text{Electrical potential energy} = \frac{kQq}{r}, \quad V_{\text{electric}} =  \frac{kQ}{r}[/math]

IV)              [math]F = qvB[/math]

  • a) Describe and explain:

  • I)  Uniform Electric Field

  • ⇒ Description:
  • A uniform electric field is one where the electric field strength E is constant in magnitude and direction.
  • Such a field is created, for example, between two parallel conducting plates connected to a voltage source.
  • ⇒ Formula:
  • [math]E = \frac{V}{d}[/math]
  • Where:
  • – E: Electric field strength (N/C or V/m),
  • – V: Potential difference between the plates (volts),
  • – d: Distance between the plates (meters).
  • Figure 1 Uniform electric field
  • ⇒ Explanation:
  • A test charge placed in the field experiences a constant force F due to the uniform E:
  • [math]F = qE[/math]
  •  Where q is the charge on the particle.
  • The field lines between the plates are parallel and equally spaced, indicating uniformity.

  • II) Electric Field of a Charged Object and Force on a Charge:

  • ⇒  Electric Field Due to a Point Charge:
  • For a charged object, the electric field at a distance r from the charge Q is given by:
  • [math]E = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r^2}[/math]
  • Where:
  • – [math]\varepsilon_0[/math]: Permittivity of free space ([math]8.85 × 10^{-12} F/m[/math]).
  • Figure 2 Electric field
  • ⇒ Force on a Charge in the Field:
  • A charge q placed in the field of Q experiences a force given by Coulomb’s law:
  • [math]F = qE \\ E = \frac{1}{4 \pi \varepsilon_0} \frac{qQ}{r^2}[/math]
  • ⇒ Inverse Square Law:
  • The electric field strength and force decrease as [math]\frac{1}{r^2}[/math], demonstrating the inverse square relationship.
  • ⇒ Field Lines:
  • Field lines originate from positive charges and terminate on negative charges.
  • The density of lines indicates the strength of the field.

  • III)  Electrical Potential Energy and Electric Potential

  • ⇒ Electric Potential Energy (U):
  • The potential energy of a charge q at a distance r from a charge Q is:
  • [math]U = \frac{1}{4 \pi \varepsilon_0} \frac{qQ}{r}[/math]
  • ⇒ Electric Potential (V):
  • The potential at a distance r from a charge Q is:
  • [math]V = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r}[/math]
  • Electric potential is a scalar quantity and represents the energy per unit charge:
  • [math]V = \frac{U}{q}[/math]
  • 1/r Relationship:
  • Both electric potential and electric potential energy follow an inverse relationship with r. This means that as distance increases, the potential and energy decrease.

  • IV. Evidence for the Discreteness of Charge on the Electron

  • ⇒ Experiment:
  • Millikan’s Oil Drop Experiment provided direct evidence of quantized charge.
  • ⇒ Description:
  • – Fine oil droplets were sprayed into a chamber and allowed to fall under gravity.
  • – A uniform electric field was applied to balance the weight of the droplets.
  • – The force on the droplets was calculated using: [math]F = qE[/math]
  • – The charges on the droplets were measured and found to be multiples of a fundamental charge([math]e = 1.6 × 10^{-19} C[/math]).
  • ⇒  Conclusion:
  • The smallest measurable charge was identified as the charge of a single electron.

  • V. Force on a Moving Charged Particle in a Uniform Magnetic Field

  • ⇒ Lorentz Force:
  • A charged particle of charge q, moving with velocity v in a magnetic field B, experiences a force given by:
  • [math]F = qvB \, sin⁡θ[/math]
  • Where:
  • – θ: Angle between v and B.
  • The force is maximum when [math]θ = 90^0[/math] (particle moving perpendicular to the field).
  • The force is zero when [math]θ = 0^0[/math](particle moving parallel to the field).
  • The force causes the particle to move in a circular or helical path, with radius:
  • [math]r = \frac{mv}{qB}[/math]

  • VI. Similarities and Differences Between Electric and Gravitational Fields

Aspect Electric Field Gravitational Field
Source Electric Charges Masses
Force Law [math]F = \frac{1}{4 \pi \varepsilon_0} \frac{q_1 q_2}{r^2}[/math] [math]F = G \frac{m_1 m_2}{r^2}[/math]
Field Strength [math]E = \frac{F}{q} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r^2}[/math] [math]g = \frac{F}{m} = \frac{G M}{r^2}[/math]
Potential [math]V = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r}[/math] [math]\phi = -\frac{G M}{r}[/math]
Attractive / Repulsive Can be both attractive and repulsive Always attractive
Field Lines Begin at positive charges, end at negative charges Always point toward the mass
  •  
  • ⇒  Similarity:
  • Both fields follow the inverse square law.
  • ⇒ Difference:
  • Electric forces can be attractive or repulsive, whereas gravitational forces are always attractive.

  • b) Make appropriate use of:

  • I. Definitions of Key Terms

  • ⇒ Charge:
  • Definition:
  • – Charge (q) is a fundamental property of matter that causes it to experience a force in an electric field.
  • Unit: Coulomb (C).
  • Types: Positive (+q) and negative (−q).
  • ⇒  Electric Field (E):
  • Definition:
  • – The region around a charged object where other charges experience a force.
  • Formula:
  • [math]E = \frac{F}{q} \\
    E = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r^2}[/math]
  • – E: Electric field strength (N/C or V/m)
  • – F: Force experienced by a test charge (q)
  • – Q: Source charge,
  • – r: Distance from the source charge.
  • Direction:
  • – Outward from positive charges, inward to negative charges.
  • ⇒ Electric Potential (V):
  • Definition:
  • – The work done per unit charge in bringing a test charge from infinity to a point in an electric field.
  • Formula:
  • [math]V = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r^2}[/math]
  • – V: Electric potential (volts),
  • – Q: Source charge,
  • – r: Distance from the source charge.
  • ⇒ Equipotential Surface:
  • Definition:
  • A surface where the electric potential is the same at every point.
  • No work is done when moving a charge along an equipotential surface
  • Equipotential surfaces are always perpendicular to electric field lines.
  • ⇒  Electron-volt (eV):
  • Definition:
  • The energy gained or lost by an electron moving through a potential difference of 1 volt.
  • Conversion:
  • [math]1eV= 1.6 × 10^{-19} J[/math]

  • II. Graphs and Their Interpretations

  • ⇒  Graph of Electric Potential vs. Distance:
  • Description:
  • – Electric potential (V) decreases with increasing distance (r) from a positive charge
  • – For a point charge:
  • [math]V(r) = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r}[/math]
  • Graph:
  • – A curve that asymptotically approaches zero as r→∞.
  • Figure 3 Graph between electric potential and distance
  • Interpretation:
  • – The area under the graph of E r represents the electric potential difference.
  • ⇒  Electric Potential Energy Change (ΔU) vs. Distance:
  • Formula:
  • [math]\Delta U = q \Delta V \\
    \Delta U = \frac{qQ}{4 \pi \varepsilon_0} \left( \frac{1}{r_1} – \frac{1}{r_2} \right)[/math]
  • Graph:
  • – A decreasing curve, as potential energy decreases with distance.
  • Figure 4 Potential energy curve
  • Interpretation:
  • – The area under the graph of F r between two points represents the change in electric potential energy (ΔU).
  • ⇒  Force and Tangent of Potential Energy Graph:
  • Relation:
  • – Force is the negative gradient of potential energy (U) vs. distance:
  • [math]F = – \frac{dU}{dr}[/math]
  • Graph:
  • – A steep negative slope indicates a strong force.
  • ⇒  Field Strength and Tangent of Potential Graph:
  • Relation:
  • – Electric field strength is the negative gradient of the electric potential (V) vs. distance:
  • [math]E = – \frac{dV}{dr}[/math]
  • Graph:
  • – A steep slope of V r indicates a strong electric field.

  • III. Equipotential Surfaces and Electric Field Diagrams

  • ⇒ Representation of Electric Field Lines:
  • Positive Point Charge:
  • – Field lines radiate outward
  • – Equipotential surfaces are concentric spheres centered on the charge.
  • Negative Point Charge:
  • – Field lines converge inward.
  • – Equipotential surfaces are concentric spheres.
  • ⇒ Uniform Electric Field:
  • Field Lines:
  • – Parallel and equally spaced.
  • Equipotential Surfaces:
  • – Planes perpendicular to the field lines.
  • ⇒  Parallel Plates:
  • Field Lines:
  • – Straight lines from the positive plate to the negative plate.
  • Equipotential Surfaces:
  • – Planes parallel to the plates.
  • ⇒  Dipole:
  • Field Lines:
  • – Originate from the positive charge and terminate on the negative charge.
  • Equipotential Surfaces:
  • – Symmetric curves around the dipole.

  • IV. Practical Uses and Applications

  • Electric Field Strength (E):
  • – Determining the force on a test charge:
  • [math]F = qE[/math]
  • Electric Potential (V):
  • – Calculating work done in moving a charge:
  • [math]W = qΔV[/math]
  • Energy Conversions:
  • – Conversion of potential energy to kinetic energy:
  • [math]\frac{1}{2} mv^2 = q∆V[/math]
  • Graphical Analysis:
  • Helps in visualizing and calculating quantities like work, potential difference, and field strength.

  • c) Make calculations and estimates involving:

  • I)  Radial Electric Force

  • ⇒ Formula:
  • [math]F_{\text{electric}} = k \frac{qQ}{r^2}[/math]
  • – [math]F_{\text{electric}} [/math]​: Force between two point charges (N),
  • – q: Test charge (C),
  • – Q: Source charge (C),
  • – r: Distance between charges (m)
  • – k: Coulomb’s constant, given by:
  • [math]k = \frac{1}{4 \pi \varepsilon_0} \\
    k = 8.99 \times 10^9 \, \text{(Nm}^2\text{/C}^2)[/math]
  • ⇒ Explanation:
  • The force is proportional to the product of the charges and inversely proportional to the square of the distance between them.
  • Direction:
  • – If q and Q are of the same sign, the force is repulsive.
  • – If q and Q are of opposite signs, the force is attractive.

  • II)  Radial Electric Field Strength:

  • ⇒  Relationship to Force:
  • [math]E_{electric} = \frac{F_{electric}}{q}[/math]
  • Substituting
  • [math]F_{\text{electric}} = k \frac{qQ}{r^2} \\
    E_{\text{electric}} = \left( k \frac{qQ}{r^2} \right) \frac{1}{q} \\
    E_{\text{electric}} = k \frac{Q}{r^2}[/math]​ ​
  • – [math]E_{\text{electric}}[/math]: Electric field strength (N/C or V/m),
  • – Q: Source charge,
  • – r: Distance from the source charge.
  • ⇒  Interpretation:
  • – The electric field strength is independent of the test charge (q).
  • – It depends only on the source charge (Q) and the distance (r).

  • III)   Electric Field from Electric Potential

  • ⇒ Radial Field:
  • [math]E_{\text{electric}} = -\frac{dV_{\text{electric}}}{dr}[/math]
  • – [math]V_{\text{electric}}[/math]​: Electric potential due to a point charge,
  • [math]V_{\text{electric}} = k \frac{Q}{r} [/math]
  • – Differentiating ​[math]V_{\text{electric}}[/math] with respect to r:
  • [math]E_{\text{electric}} = -\frac{d}{dr} \left( k \frac{Q}{r} \right) \\
    E_{\text{electric}} = k \frac{Q}{r^2}[/math]
  • This confirms that the electric field is the negative gradient of the electric potential.
  • ⇒  Uniform Field:
  • [math]E_{electric} = \frac{V_{electric}}{d}[/math]
  • – d: Distance between two points in the uniform electric field.

  • IV)  Electrical Potential Energy

  • ⇒  Formula:
  • [math]U_{electric} = k \frac{Qq}{r}[/math]
  • – [math]U_{electric}[/math]​: Electrical potential energy (J),
  • – Q: Source charge,
  • – q: Test charge,
  • – r: Distance between charges.
  • ⇒  Interpretation:
  • Electrical potential energy decreases as the charges are moved farther apart (r increases).
  • If Q and q have the same sign,
  • [math]U_{electric} > 0[/math]; for opposite signs, [math]U_{electric} < 0[/math].
  • V)  Magnetic Force on a Moving Charge
  • ⇒  Formula:
  • [math]F = qvB[/math]
  • – F: Magnetic force (N),
  • – q: Charge of the particle (C),
  • – v: Velocity of the particle perpendicular to the magnetic field (m/s)
  • – B: Magnetic flux density (T, teslas).
  • Interpretation:
  • – The force is maximum when the charge moves perpendicular to the magnetic field ([math]θ = 90^0).[/math]).
  • – The force is zero when the charge moves parallel to the field ([math]θ = 0^0[/math]).
  • ⇒  Direction (Right-Hand Rule):
  • Thumb: Direction of velocity (v),
  • Fingers: Direction of magnetic field (B),
  • Palm: Direction of force for positive charges (opposite for negative charges).

  • VI)  Examples and Applications

  • ⇒  Example 1: Electric Force Calculation
  • Two charges, Q=2.0 and q=1.0 , are separated by r=0.1 m. Find the force between them.
  • [math]F_{\text{electric}} = k \frac{qQ}{r^2} \\
    F_{\text{electric}} = \frac{(8.99 \times 10^9) \times \left( (2 \times 10^{-6}) (1 \times 10^{-6}) \right)}{(0.1)^2} \\
    F_{\text{electric}} = 1.798 \, \text{N}[/math]
  • ⇒  Example 2: Electric Potential Energy
  • Using the same charges as above, calculate the potential energy at r=1m.
  • [math]U_{\text{electric}} = k \frac{Qq}{r} \\
    U_{\text{electric}} = \frac{(8.99 \times 10^9) \times \left( (2 \times 10^{-6}) (1 \times 10^{-6}) \right)}{0.1} \\
    U_{\text{electric}} = 0.1798 \, \text{J}[/math]
  • ⇒  Example 3: Magnetic Force
  • A proton (q= [math]1.6 × 10^{-19} C[/math]) moves with a velocity [math]v = 1 × 10^6 m/s[/math] perpendicular to a magnetic field B=0.01 T. Find the force on the proton.
  • [math]F = q v B \\
    F = (1.6 \times 10^{-19}) \times (1 \times 10^6) \times (0.01) \\
    F = 1.6 \times 10^{-15} \, \text{N}[/math]
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