Electric Charges & Fields: Gauss's Law

The magnitude of the larger electric field.

The average of the two electric field magnitudes.

The vector sum of $\vec{E}_1$ and $\vec{E}_2$.

The product of the two electric field magnitudes.

Constant and non-zero.

Increasing linearly with distance from the center.

Zero.

Decreasing exponentially with distance from the center.

$E \propto \lambda r$

$E \propto \frac{\lambda}{r^2}$

$E \propto \frac{\lambda}{r}$

$E \propto \lambda r^2$

The electric field vector.

The permittivity of free space.

A tiny bit of charge.

The distance from the charge to the point where the field is measured.

Direct integration of $dq/r^2$ without considering components.

Integration of the perpendicular components of the electric field.

Integration of the x-component of the electric field, using the distance from each dq to the point.

No integration is needed; the electric field is zero.

Tangential to the surface of the sphere.

Radially outward from the center of the sphere (or inward for negative charges).

Parallel to the surface of the sphere.

Zero.

As a constant value.

In terms of the total charge Q divided by the length L.

As $\lambda(x) dx$, where $\lambda(x)$ is the linear charge density as a function of position x.

Using only the distance r from the charge to the point P.

The electric field is radial, and you integrate the charge density directly.

The electric field is tangential, and you integrate the charge density directly.

Use symmetry to determine the direction (e.g., along the y-axis if the arc is symmetric about the y-axis), and then set up an integral to find the component of the electric field in that direction.

The electric field is zero due to symmetry; no integration is needed.

The symmetry of the problem makes the integral trivial.

Expressing $dq$ in terms of $dx$ and the linear charge density.

Determining the limits of integration.

Correctly expressing the distance $r$ from $dq$ to the point and integrating the components of the electric field.