Outside the sphere, the electric field decreases proportionally to the inverse square of the radius.

The electric field increases linearly with the distance from the center until r = R (the radius of the sphere).

The electric potential is constant inside the conductor.

The electric field is zero.

1. Break the total charge Q into tiny pieces dq. 2. Find the electric field dE due to each piece. 3. Integrate dE to find the total electric field E.

1. Choose a Gaussian surface. 2. Calculate the electric flux through the Gaussian surface. 3. Calculate the enclosed charge. 4. Apply Gauss' Law: $$\oint E \cdot dA = \frac{Q_{enc}}{ε_0}$$

1. Use Gauss' Law to find an expression for E. 2. Plug that into: $$ΔV = -\int E \cdot dl$$

1. Express dq in terms of λ and dl. 2. Find dE using $$dE = k \frac{dq}{r^2}$$. 3. Integrate dE to find the total electric field E.

1. Express dq in terms of dy using the linear charge density λ. 2. Use the Pythagorean theorem to find r. 3. Integrate to find the total electric field.

1: Radius of the ring (a), 2: Distance from the center of the ring along the axis (x), 3: Radius vector (r), 4: Charge element (dq).

1: Length of the line of charge (L), 2: Distance from the line of charge (x), 3: Position along the line of charge (y), 4: Charge element (dq).

1: Radius of the sphere (R), 2: Gaussian surface (sphere of radius r), 3: Distance from the center (r).

1: Insulating sheet, 2: Gaussian surface (rectangle), 3: Area (A).