4.2 Calculating electric potential from electric field

Electric potential and electric field are closely linked in electromagnetism. We can calculate potential from field using line integrals, which measure change along a path. This connection helps us understand energy in electric systems.

The relationship between potential and field is key to solving many electrostatics problems. By integrating the field, we can find potential differences and determine the work done on charges in electric fields.

Scalar and Vector Fields

Scalar Fields and Vector Fields

• Scalar field assigns a single value (scalar) to each point in space
• Common scalar fields include temperature, pressure, and electric potential
• Vector field assigns a vector to each point in space
• Vector has both magnitude and direction
• Common vector fields include electric field, magnetic field, and velocity field of a fluid

Gradient of a Scalar Field

• Gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field
• Magnitude of the gradient vector equals the rate of change of the scalar field in that direction
• Gradient operator $\nabla$ is used to calculate the gradient
  • In Cartesian coordinates: $\nabla = \hat{i}\frac{\partial}{\partial x} + \hat{j}\frac{\partial}{\partial y} + \hat{k}\frac{\partial}{\partial z}$
• The gradient of a scalar field $\phi$ is denoted as $\nabla \phi$
  • $\nabla \phi = \frac{\partial \phi}{\partial x}\hat{i} + \frac{\partial \phi}{\partial y}\hat{j} + \frac{\partial \phi}{\partial z}\hat{k}$

Line Integrals and Potential Difference

Line Integrals

• Line integral calculates the integral of a function along a curve or path
• For a scalar field $f(x, y, z)$ and a parametric curve $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}$, the line integral is:
  • $\int_C f(x, y, z) ds = \int_a^b f(\vec{r}(t)) |\vec{r}'(t)| dt$
• For a vector field $\vec{F}(x, y, z)$ and a parametric curve $\vec{r}(t)$, the line integral is:
  • $\int_C \vec{F} \cdot d\vec{r} = \int_a^b \vec{F}(\vec{r}(t)) \cdot \vec{r}'(t) dt$

Potential Difference and Path Independence

• Potential difference $\Delta V$ between two points $A$ and $B$ in an electric field $\vec{E}$ is the line integral of $\vec{E}$ from $A$ to $B$:
  • $\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{r}$
• If the electric field is conservative (i.e., electrostatic field), the potential difference is independent of the path taken between $A$ and $B$
• Path independence implies that the line integral of $\vec{E}$ around any closed loop is zero:
  • $\oint \vec{E} \cdot d\vec{r} = 0$

Superposition Principle

Superposition of Electric Fields and Potentials

• Superposition principle states that the total electric field or potential at a point due to multiple sources is the vector sum of the individual fields or potentials
• For electric fields: $\vec{E}_{total} = \vec{E}_1 + \vec{E}_2 + ... + \vec{E}_n$
  • Example: The electric field at a point due to two point charges is the vector sum of the fields due to each charge individually
• For electric potentials: $V_{total} = V_1 + V_2 + ... + V_n$
  • Example: The electric potential at a point due to multiple point charges is the sum of the potentials due to each charge individually
• Superposition principle is valid because Maxwell's equations are linear
• Allows for the calculation of complex electric fields and potentials by breaking them down into simpler components