Electric Fields and Acceleration of Charges

Electric Field and Electric Force Concepts

• Electric Field (E): A region of space around a charged object where another charged object will experience a force.

• Electric Field Strength (E): The force per unit positive charge experienced at a point in an electric field.

  • Units: N/C (Newtons per Coulomb) or V/m (Volts per meter)
  • Uniform Electric Field: An electric field with constant magnitude and direction. Created between two parallel charged plates.

  • Electric Potential Difference (V): The work done per unit charge to move a positive charge between two points in an electric field.

  • Units: Volts (V)

Formulas

• Electric Field between parallel plates:
  $$E = \frac{V}{d}$$
  Where:

  • $E$ = Electric field strength (V/m)
  • $V$ = Potential difference between the plates (V)
  • $d$ = Distance between the plates (m)

• Electric Force (F): The force experienced by a charge in an electric field.
  $$F = qE$$
  Where:

  • $F$ = Electric force (N)
  • $q$ = Charge (C)
  • $E$ = Electric field strength (N/C or V/m)

KEY TAKEAWAY: Electric field strength is the voltage divided by the distance between the plates in a uniform electric field. The electric force on a charge is the product of the charge and the electric field strength.

Potential Energy Changes in a Uniform Electric Field

• Work Done (W): The energy transferred when a force moves an object over a distance. In the context of electric fields, it’s the energy required to move a charge against the electric force.
• Electric Potential Energy (U): The potential energy a charge possesses due to its position in an electric field.

Relationship between Work and Potential Energy

• The work done to move a charge in an electric field is equal to the change in its electric potential energy.
• If a positive charge moves in the direction of the electric field (from high to low potential), its potential energy decreases, and the work done is positive.
• If a positive charge moves against the direction of the electric field (from low to high potential), its potential energy increases, and the work done is negative.

Formulas

• Work Done in a Uniform Electric Field:
  $$W = qV$$
  Where:

  • $W$ = Work done (J)
  • $q$ = Charge (C)
  • $V$ = Potential difference (V)

• Relating Work and Kinetic Energy: If the work done on a charged particle is the only work done, then it is equal to the change in kinetic energy.
  $$W = \Delta KE = KE_f - KE_i$$
  $$qV = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$
  Where:

  • $m$ = mass of the particle (kg)
  • $v_f$ = final velocity (m/s)
  • $v_i$ = initial velocity (m/s)

  If a charged particle starts from rest ($v_i = 0$):
  $$\frac{1}{2}mv^2 = qV$$

EXAM TIP: When calculating the work done, ensure you use the correct sign for the charge. Electrons have a negative charge, which can affect the sign of the work done.

Magnitude of the Force on a Charged Particle in a Uniform Electric Field

• Force on a Charged Particle: As mentioned above, the electric force on a charged particle in an electric field is given by $F = qE$.
• Acceleration: The electric force causes the charged particle to accelerate. Using Newton’s Second Law ($F = ma$), we can find the acceleration:

Formulas

• Force:
  $$F = qE$$

• Acceleration:
  $$a = \frac{F}{m} = \frac{qE}{m}$$
  Where:

  • $a$ = Acceleration (m/s²)
  • $m$ = Mass of the particle (kg)

Example: Electron Acceleration

Consider an electron (charge $q = -1.6 \times 10^{-19} C$, mass $m = 9.11 \times 10^{-31} kg$) placed in an electric field of strength $E = 1000 N/C$.

1. Force: $F = qE = (-1.6 \times 10^{-19} C)(1000 N/C) = -1.6 \times 10^{-16} N$. The negative sign indicates the force is in the opposite direction to the electric field (since the electron is negatively charged).
2. Acceleration: $a = \frac{F}{m} = \frac{-1.6 \times 10^{-16} N}{9.11 \times 10^{-31} kg} = -1.76 \times 10^{14} m/s^2$. This is a very large acceleration.

COMMON MISTAKE: Forgetting to include the correct sign for the charge when calculating force and acceleration. This will impact the direction of the force and acceleration.

Applications

• Electron Guns: Used in cathode ray tubes (CRTs), oscilloscopes, and electron microscopes to accelerate electrons to high speeds.
• Particle Accelerators: Devices like linear accelerators (LINACs) and cyclotrons use electric fields to accelerate charged particles to very high energies for research purposes.
• Inkjet Printers: Use electric fields to direct charged ink droplets onto paper to form images.

STUDY HINT: Practice solving problems involving electric fields, forces, and potential energy. Draw diagrams to visualize the electric field and the motion of the charged particles.

Summary Table

VCAA FOCUS: Pay close attention to problems involving energy transformations and the relationship between electric potential energy and kinetic energy. VCAA often includes questions that require you to apply multiple concepts to solve a problem.