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Electric Field The electric field vector 𝐄 is the negative...

Electric Field The electric field vector $\mathbf{E}$ is the negative of the gradient of the electrostatic potential $V$. That is, $\mathbf{E}=-\nabla V$. If $Q$ is a point charge at the origin, then the electrostatic potential $V$ at the point $P(x, y, z)$ is given by $V=\frac{k Q}{\sqrt{x^{2}+y^{2}+z^{2}}},$ where $k$ is a constant. (a) Find the electric field vector $\mathbf{E}$ at the point $P(x, y, z)$. (b) Use (a) to show that the magnitude of the electric field is $\|\mathbf{E}\|=\frac{k Q}{r^{2}},$ where $r$ is the distance from the origin to the point $P(x, y, z)$ (c) Verify that the electric field vector $\mathbf{E}$ is normal to the equipotential surfaces (surfaces on which the electrostatic potential $V$ is constant).

Electric Field The electric field vector $\mathbf{E}$ is the negative of the gradient of the electrostatic potential $V$. That is, $\mathbf{E}=-\nabla V$. If $Q$ is a point charge at the origin, then the electrostatic potential $V$ at the point $P(x, y, z)$ is given by $V=\frac{k Q}{\sqrt{x^{2}+y^{2}+z^{2}}},$ where $k$ is a constant.
(a) Find the electric field vector $\mathbf{E}$ at the point $P(x, y, z)$.
(b) Use (a) to show that the magnitude of the electric field is $\|\mathbf{E}\|=\frac{k Q}{r^{2}},$ where $r$ is the distance from the origin to the point $P(x, y, z)$
(c) Verify that the electric field vector $\mathbf{E}$ is normal to the equipotential surfaces (surfaces on which the electrostatic potential $V$ is constant).

Key Concepts

Electrostatic Potential

Electrostatic potential is a scalar quantity that represents the potential energy per unit charge at any point in space due to a distribution of charges. It is a fundamental concept in electrostatics, serving as the source from which the electric field is derived. The potential facilitates the calculation of work done by or against the field when moving a charge from one point to another.

Electric Field as the Negative Gradient of Potential

The electric field is defined as the negative gradient of the electrostatic potential, establishing a direct relationship between a scalar potential field and a vector field. This means that the electric field points in the direction of the steepest decrease in potential, and its magnitude reflects how rapidly the potential changes, a principle that underpins much of classical electromagnetism.

Gradient Operator

The gradient operator is a vector differential operator that, when applied to a scalar field like the electrostatic potential, provides a vector field indicating the direction and rate of the highest increase at each point. The negative gradient is particularly important in electrostatics as it gives the corresponding electric field, describing both the magnitude and the direction of the force exerted on a charge.

Coulomb's Law and Point Charges

Coulomb's Law characterizes the interaction between point charges and is the foundation for the potential and field distributions around such charges. It states that the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of their separation distance. This law allows the derivation of the potential created by a single point charge, which then leads to the corresponding electric field through differentiation.

Equipotential Surfaces

Equipotential surfaces are surfaces over which the electrostatic potential remains constant. They are significant because the electric field is always perpendicular to these surfaces, reflecting that no work is done in moving a charge along an equipotential surface. This perpendicularity provides insight into the field's geometry and is essential for understanding energy conservation and field lines in electrostatics.

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