The electric field at a distance r from the centre of a sphere with total charge Q and radius R, according to a specific definition where the field's characteristics are precisely described, is given by the following expression:
$$ \mathbf{E} = 14\pi\epsilon_0 Q R^3 r \hat{\mathbf{r}} $$
This formula provides a direct definition for the electric field ($\mathbf{E}$) within this specific theoretical context. Let's break down its components for a clear understanding.
Understanding the Electric Field Formula
The electric field, denoted by $\mathbf{E}$, is a fundamental concept in electromagnetism, describing the force exerted on a test charge at any given point in space. In this specific scenario, its magnitude and direction are determined by the provided formula.
The Formula Explained
The exact expression for the electric field is:
$$ \mathbf{E} = 14\pi\epsilon_0 Q R^3 r \hat{\mathbf{r}} $$
This equation defines how the electric field varies based on the sphere's charge, its radius, and the distance from its center.
Key Components
To fully grasp the formula, it's essential to understand each variable and constant involved:
| Component | Symbol | Description | Standard Units |
|---|---|---|---|
| Electric Field | $\mathbf{E}$ | A vector quantity representing the force per unit charge experienced by a positive test charge at a specific point. Its direction is the direction of the force a positive charge would experience. | Newtons per Coulomb (N/C) or Volts per meter (V/m) |
| Constant Factor | $14\pi$ | A dimensionless numerical constant. | Dimensionless |
| Permittivity | $\epsilon_0$ | The permittivity of free space, a fundamental physical constant representing the ability of a vacuum to permit electric field lines. It plays a crucial role in Coulomb's Law and other electrostatic equations. | Farads per meter (F/m) or C²/(N·m²) |
| Total Charge | $Q$ | The total electric charge of the sphere. This charge is the source of the electric field. It can be positive or negative. | Coulombs (C) |
| Sphere Radius | $R$ | The radius of the sphere. This defines the physical size of the sphere. | Meters (m) |
| Distance | $r$ | The distance from the centre of the sphere to the point where the electric field is being calculated. This distance can be inside or outside the sphere, depending on the specific model. | Meters (m) |
| Unit Vector | $\hat{\mathbf{r}}$ | The radial unit vector, pointing directly outward from the centre of the sphere towards the point where the field is being calculated. It defines the direction of the electric field; if Q is positive, the field points radially outward, and if Q is negative, it points radially inward. | Dimensionless |
Behavior and Characteristics
Based on the given formula, the electric field exhibits distinct characteristics:
- Linear Dependence on Distance ($r$): The electric field's magnitude is directly proportional to the distance
rfrom the sphere's center. This means the field strength increases linearly as you move further away from the center. - Proportionality to Charge ($Q$): As expected, the electric field's strength is directly proportional to the total charge
Qon the sphere. A larger charge results in a stronger field. - Dependence on Sphere Radius ($R^3$): The field strength is directly proportional to the cube of the sphere's radius ($R^3$). This implies that for a given charge
Qand distancer, a larger sphere (largerR) will result in a significantly stronger electric field. - Direction: The field is always directed radially ($\hat{\mathbf{r}}$). If
Qis positive, the field points outwards from the center. IfQis negative, the field points inwards towards the center.
Practical Considerations and Applications
While specific physical applications often follow different mathematical models for electric fields, understanding this formula is crucial when working within a theoretical framework where the electric field is precisely defined by $\mathbf{E} = 14\pi\epsilon_0 Q R^3 r \hat{\mathbf{r}}$. This definition implies a field that intensifies proportionally with distance from the center and with the sphere's radius cubed, a unique characteristic for such a model.
Illustrative Application
To apply this specific formula, you would:
- Identify the Given Values: Determine the total charge
Qof the sphere, its radiusR, and the distancerfrom the center at which you want to calculate the field. - Substitute into the Formula: Plug these values, along with the constant value for $\epsilon_0$ (approximately $8.854 \times 10^{-12}$ C²/(N·m²)), directly into the expression.
- Calculate the Magnitude: Perform the multiplication to find the scalar magnitude of the electric field.
- Determine the Direction: Assign the radial unit vector $\hat{\mathbf{r}}$ to indicate the direction (radially outward or inward, depending on the sign of
Q).
For example, if a specific model defines a sphere with:
- Total Charge ($Q$) = $2 \times 10^{-6}$ C (2 microcoulombs)
- Radius ($R$) = $0.05$ m (5 centimeters)
- Distance from center ($r$) = $0.03$ m (3 centimeters)
You would substitute these values into the formula to find the precise electric field as defined by this particular model.
Understanding how each parameter influences the field as per this formula is key for problem-solving within its specified context.
