Gravitational and Electric Fields: Point Mass/Charge Fields

Introduction to Fields

• A field is a region of space where an object experiences a force.
• Gravitational fields exert forces on objects with mass.
• Electric fields exert forces on objects with electric charge.
• We consider fields created by point masses (gravity) or point charges (electricity).

KEY TAKEAWAY: Fields are models to explain forces acting at a distance.

Gravitational Fields

Direction of the Gravitational Field

• The direction of the gravitational field is the direction of the force that would be exerted on a test mass placed in the field.
• Gravitational force is always attractive.
• The gravitational field lines point radially inwards towards the point mass.

Shape of the Gravitational Field

• The gravitational field around a point mass is radial.
• Field lines are closest together near the point mass, indicating a stronger field.
• As distance from the point mass increases, the field lines spread out, indicating a weaker field.

Magnitude of the Gravitational Field

• The magnitude of the gravitational field, $g$, at a distance $r$ from a point mass $M$ is given by Newton’s Law of Universal Gravitation:

  $$g = \frac{GM}{r^2}$$

  Where:
  * $G$ is the gravitational constant (\$6.674 \times 10^{-11} \, Nm^2kg^{-2}$)
  * $M$ is the mass of the object creating the field
  * $r$ is the distance from the center of the mass

• Inverse Square Law: The gravitational field strength is inversely proportional to the square of the distance from the point mass. This means doubling the distance reduces the field strength by a factor of four.

Gravitational Potential Energy Changes

• Gravitational potential energy ($E_g$) is the energy an object possesses due to its position in a gravitational field.

• For a point mass $m$ in the gravitational field of another point mass $M$, the change in gravitational potential energy ($\Delta E_g$) when moving from distance $r_1$ to $r_2$ is:

  $$\Delta E_g = -GMm \left(\frac{1}{r_2} - \frac{1}{r_1}\right)$$

• Qualitative Analysis:

  • As a mass moves away from another mass (against the gravitational field), its gravitational potential energy increases. Work is done on the mass by an external force.
  • As a mass moves towards another mass (along the gravitational field), its gravitational potential energy decreases. Work is done by the gravitational field.

EXAM TIP: Always consider the direction of movement relative to the field when determining potential energy changes.

Electric Fields

Direction of the Electric Field

• The direction of the electric field is the direction of the force that would be exerted on a positive test charge placed in the field.
• Electric force can be attractive or repulsive, depending on the charges involved.
• Field lines point:
  • Away from positive charges
  • Towards negative charges

Shape of the Electric Field

• The electric field around a point charge is radial.
• Field lines are closest together near the point charge, indicating a stronger field.
• As distance from the point charge increases, the field lines spread out, indicating a weaker field.
• For multiple charges, the field shape becomes more complex, representing the vector sum of the individual fields.

Magnitude of the Electric Field

• The magnitude of the electric field, $E$, at a distance $r$ from a point charge $Q$ is given by Coulomb’s Law:

  $$E = \frac{kQ}{r^2}$$

  Where:
  * $k$ is Coulomb’s constant (\$8.988 \times 10^9 \, Nm^2C^{-2}$)
  * $Q$ is the magnitude of the charge creating the field
  * $r$ is the distance from the center of the charge

• Inverse Square Law: The electric field strength is inversely proportional to the square of the distance from the point charge.

Electric Potential Energy Changes

• Electric potential energy ($E_e$) is the energy a charge possesses due to its position in an electric field.

• For a charge $q$ in the electric field of another charge $Q$, the change in electric potential energy ($\Delta E_e$) when moving from distance $r_1$ to $r_2$ is:

  $$\Delta E_e = kQq \left(\frac{1}{r_2} - \frac{1}{r_1}\right)$$

• Qualitative Analysis:

  • If $q$ and $Q$ have the same sign (both positive or both negative):
    • As $q$ moves away from $Q$ (against the electric field), its electric potential energy increases.
    • As $q$ moves towards $Q$ (along the electric field), its electric potential energy decreases.
  • If $q$ and $Q$ have opposite signs:
    • As $q$ moves away from $Q$ (against the electric field), its electric potential energy decreases.
    • As $q$ moves towards $Q$ (along the electric field), its electric potential energy increases.

COMMON MISTAKE: Forgetting to consider the signs of the charges when determining electric potential energy changes. Like charges repel, opposite charges attract.

Comparison of Gravitational and Electric Fields

STUDY HINT: Create a table summarizing the similarities and differences between gravitational and electric fields.

Uniform vs. Non-Uniform Fields

• Uniform Field: The field strength and direction are constant throughout the region of space. Electric fields between charged parallel plates are a good approximation of a uniform field. Gravitational field near earth’s surface is often treated as uniform.
• Non-Uniform Field: The field strength and/or direction vary with position. Radial fields around point masses and point charges are non-uniform.

VCAA FOCUS: VCAA often requires calculations involving both uniform and non-uniform fields. Make sure you understand when and how to apply the relevant formulas.

Applications

• Gravitational Fields: Satellite motion, planetary orbits, projectile motion.
• Electric Fields: Particle accelerators, cathode ray tubes (CRTs), inkjet printers.

APPLICATION: Particle accelerators use electric fields to accelerate charged particles to very high speeds for research purposes.