Potential near a point charge B + What is the potential difference between A and B?

+ A B Change in potential along a short section of the path: Now integrate along the path:

Change in potential along a short section of the path: + B

Potential difference near a point charge B A +

Potential at one location Let rA go to infinity… B +

Potential at one location The potential at a distance r from a point charge, relative to infinity: +

Potential at one location The potential at a distance r from a point charge, relative to infinity: + +

Potential energy of two charges The potential energy of two point charges, relative to infinity: + +

Potential energy of a system of charges q1 q2 q3

Finding the field from the potential The change in potential along a very small path:

Finding the field from the potential Choose a path that only goes in the x-direction (dy = dz = 0):

Finding the field from the potential Choose a path that only goes in the x-direction (dy = dz = 0): (holding y and z fixed)

Finding the field from the potential Choose a path that only goes in the x-direction (dy = dz = 0): (partial derivative)

Finding the field from the potential Choose a path that only goes in the y-direction (dx = dz = 0):

Finding the field from the potential Choose a path that only goes in the z-direction (dx = dy = 0):

Electric field is the negative gradient (梯度) of the potential

Electric field is the negative gradient (梯度) of the potential

The potential is like the height of the hill. The field is like the slope of the hill. Just remember: - positive charges go down the hill - negative charges go up!

Field around a point charge The potential near a point charge, relative to infinity: + The field strength is the gradient of the potential:

Field around a point charge The potential near a point charge, relative to infinity: + The field strength is the gradient of the potential:

Potential along the axis of a ring Potential obeys the superposition principle, just like the field. Potential due to one small piece:

Potential along the axis of a ring Potential obeys the superposition principle, just like the field. Integrate:

Potential along the axis of a ring Potential obeys the superposition principle, just like the field. Integrate:

Field along the axis of a ring The strength of the field is the negative of the potential gradient:

Field along the axis of a ring The strength of the field is the negative of the potential gradient:

Field along the axis of a ring We already calculated this field the hard way. It is often easier to first calculate the potential, then use its gradient to get the field.

The same is true for the potential. Potential due to a uniformly charged sphere + + + Remember: The field outside a charged sphere is the same as the field of a point charge. The same is true for the potential. + + Q + + + + + + +

Potential due to a uniformly charged sphere + + + + + Q + + + + + + + V(∞) = 0

Potential at the surface + + + + + Q + + + + + + + V(∞) = 0

Potential at the surface Define the surface charge density: + + + + + Q + + + + + + + V(∞) = 0

Surface potential of a protein Arabidopsis thaliana Positive (+) Negative (-) Atserpin 1

Potential in a conductor B At equilibrium, the field inside the conductor must be zero. +

Potential in a conductor So the potential inside a conductor (and at the surface) must be constant. +

Example: A negatively charged metal sphere