# Electric vs Gravitational Fields ## Field Definitions - A **field** can be defined as: - A region in which an object will experience a force, such as gravitational or electrostatic, at a distance

A gravitational field is the gravitational force per unit mass exerted on a point mass

An electric field is the electric force per unit charge exerted on a small positive test charge

- Both fields can be described in terms of **field strength**, which is defined as: $$\text{Field strength} = \frac{\text{force acting on a test object}}{\text{size of test object}}$$ - Electric field strength $E$ and gravitational field strength $g$ have very similar equations - Despite some differences, they are analogous to one another in many ways ### Nature of Test Objects - In both cases, the nature of the test object is as follows: | Field Type | Test Object | |-----------|-------------| | Gravitational fields | Small mass $m$ | | Electrostatic fields | Small positive charge $q$ | ## Uniform Fields ### Gravitational Field Strength - A gravitational field is a region of space in which objects with mass will experience a force - The gravitational field strength can be calculated using the equation: $$g = \frac{F}{m}$$ - Where: - $g$ = gravitational field strength (N kg$^{-1}$) - $F$ = gravitational force on the mass (N) - $m$ = mass (kg) - The direction of the gravitational field is always directed **towards** the centre of the mass - Gravitational forces are **always attractive** and cannot be repulsive ### Electric Field Strength - An electric field is a region of space in which an electric charge will experience a force - The electric field strength can be calculated using the equation: $$E = \frac{F}{Q}$$ - Where: - $E$ = electric field strength (N C$^{-1}$) - $F$ = electrostatic force on the charge (N) - $Q$ = charge (C) - It is important to use a positive test charge in this definition, as this determines the direction of the electric field - The electric field strength is a **vector** quantity, it is always directed: - **Away** from a positive charge - **Towards** a negative charge - Opposite charges (positive and negative) **attract** each other - Like charges (positive-positive or negative-negative) **repel** each other ## Radial Fields ### Point Charges and Point Masses - A point charge or mass produces a **radial** field - A charged sphere acts as a point charge - A spherical mass acts as a point mass - Radial fields always have an **inverse square law** relationship with distance - This means the field strength decreases by a factor of **four** when the distance $r$ is **doubled** - The graph below illustrates how both gravitational and electric field strengths follow the same inverse square law pattern: - Both decrease rapidly with distance - Both follow the $1/r^2$ relationship - Doubling the distance reduces the field strength to one quarter of its original value

### Gravitational Force Between Two Masses - The gravitational force $F_G$ between two masses is defined by: $$F_G = \frac{Gm_1m_2}{r^2}$$ - Where: - $F_G$ = gravitational force between two masses (N) - $G$ = Newton's gravitational constant (N m$^2$ kg$^{-2}$) - $m_1$, $m_2$ = two point masses (kg) - $r$ = distance between the centre of the two masses (m) ### Electric Field Strength of a Point Charge - The electric field strength $E$ at a distance $r$ due to a point charge $Q$ in free space is defined by: $$E = \frac{Q}{4\pi\varepsilon_0 r^2}$$ - Where: - $Q$ = the point charge producing the radial electric field (C) - $r$ = distance from the centre of the charge (m) - $\varepsilon_0$ = permittivity of free space (F m$^{-1}$) = $8.85 \times 10^{-12}$ F m$^{-1}$ - This equation shows: - The electric field strength in a radial field is **not constant** - As the distance $r$ from the charge increases, $E$ decreases by a factor of $1/r^2$ ## Comparison of Gravitational and Electric Fields | Property | Gravitational Fields | Electric Fields | |----------|---------------------|-----------------| | **Origin of the force** | Mass | Charge | | **Force between two point masses/charges** | $F_G = \dfrac{Gm_1m_2}{r^2}$ | $F_E = \dfrac{Q_1Q_2}{4\pi\varepsilon_0 r^2}$ | | **Type of Force** | Attractive force | Attractive force (opposite charges)
Repulsive force (like charges) | | **Field Strength** | $g = \dfrac{F}{m}$ | $E = \dfrac{F}{Q}$ | | **Field strength due to a point mass/charge** | $g = \dfrac{GM}{r^2}$ | $E = \dfrac{Q}{4\pi\varepsilon_0 r^2}$ | | **Field Lines** | Around a point mass:
- Field lines directed inwards

In a uniform field (surface of a planet):
- Field lines equally spaced and parallel | Around a (negative) point charge:
- Field lines directed inwards

In a uniform field (between charged parallel plates):
- Field lines equally spaced and parallel | | **Potential** | $V = -\dfrac{GM}{r}$ | $V = \dfrac{Q}{4\pi\varepsilon_0 r}$ | | **Equipotential Surfaces** | Around a point mass:
- Spherical equipotential surfaces

In a uniform field (surface of a planet):
- Equally spaced parallel lines | Around a point charge:
- Spherical equipotential surfaces

In a uniform field (between charged parallel plates):
- Equally spaced parallel lines | | **Work Done on a Mass or Charge** | $\Delta W = M\Delta V$ | $\Delta W = Q\Delta V$ |

Both gravitational and electrostatic forces follow an inverse square law with distance: F ∝ 1 / r².

## Key Similarities - The magnitude of the gravitational and electrostatic force between two point masses or charges are **inverse square law** relationships - The field lines around a **point mass** and **negative point charge** are identical - Both are directed radially inwards - The field lines in a **uniform** gravitational and electric field are identical - Both are equally spaced and parallel - The **gravitational field strength** and **electric field strength** both have a $1/r^2$ relationship in a radial field - The **gravitational potential** and **electric potential** both have a $1/r$ relationship - **Equipotential surfaces** for both gravitational and electric fields are: - Spherical around a point mass or charge - Equally spaced parallel lines in uniform fields - The **work done** in each field is either the product of the mass and change in potential or charge and change in potential ## Key Differences

Gravitational forces are always attractive, whilst electrostatic forces can be attractive or repulsive

- The gravitational force acts on particles with **mass** whilst the electrostatic force acts on particles with **charge** - The gravitational force is **always attractive** whilst the electrostatic force can be **attractive or repulsive** - Like charges repel, opposite charges attract - The gravitational potential is **always negative** whilst the electric potential can be either **negative or positive** - Electric potential is positive for positive charges and negative for negative charges - Gravitational potential is always negative because gravitational force is always attractive