In the previous chapter we learned about the use of the electric field concept to describe electric forces between charges. In this chapter, we'll learn about the electric potential (also called "voltage"), which gives us an alternative, complementary, and useful way to describe electric fields.

Think back to topographical maps, which you might have studied before. Here's an example:

As we discussed in class, here are the key properties of the topographical map:

Counter lines represent points of the landscape that have the same height; they also have the same gravitational potential, so we can call them equipotential lines, or equipotentials for short.

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If the landscape were perfectly smooth and frictionless, a small particle released on the landscape would experience a force that is perpendicular to the counter line at the particle's position.

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The landscape is steepest where the contour lines are closest together (think rise-over run, which is a way to calculate the slope of the landscape in a certain direction). The landscape is less steep where the contour lines are farther apart.

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Also as we discussed in class, for conservative force fields (such as electric fields), one

1P22/1P92 Problems (2011) Chapter 21 Electric PotentialFriday, January 14, 2011

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can set up a potential function (analogous to gravitational potential) that shares many of the properties of contour lines in topographical maps. Translating these properties into the realm of electric potential functions, here are their key properties:

Equipotentials represent points in space where the electric potential is constant. (In three dimensions, the equipotentials are surfaces, not lines.)

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The force experienced by a small charged particle is perpendicular to the equipotential at the particle's position. Another way to say this is that at each point in space where the electric field is NOT zero, the electric field vector is perpendicular to the equipotential at that point.

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The electric field is strongest where the equipotentials are close together; the electric field is weaker where the equipotentials are farther apart.

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CP 3 It takes 3.0 µ J of work to move a 15 nC charge from

point A to point B. It takes 5.0 µ J of work to move the same charge from C to B. What is the potential difference

VC VA?

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CP 7 A proton has been accelerated from rest through a potential difference

of 1000 V. Calculate its kinetic energy in (a) electron volts, and (b) joules. (c.) Calculate its final speed.

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CP 46 What is the potential difference

V34 in the figure?

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CP 48 At a distance r from a point charge, the electric

potential is 3000 V and the magnitude of the electric field is 2.0 × 105 V/m. (a) Calculate the distance r. (b) Calculate the electric potential and the magnitude of the electric field at a distance r/2 from the point charge.

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CP 52 A +3.0 nC charge is at x = 0 cm and a 1.0 nC charge is at x = 4 cm. At which point or points along the x-axis is the electric potential zero?

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A charge of 10 nC is placed on a metal sphere that has a radius of 5 cm. Calculate the electric potential at the following distances from the centre of the sphere: (a) 10 cm (b) 2 cm (c.) 3 cm (d) 5

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cm.

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CP 56 A glass bead with a diameter of 2.0 mm is positively charged. The potential difference between a point 2.0 mm

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charged. The potential difference between a point 2.0 mm from the surface of the bead and a point 4.0 mm from the surface of the bead is 500 V. What is the charge on the bead?

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CP 30 A switch that connects a battery to a 10 µF capacitor is closed. Several seconds later the capacitor plates are charged to ±30 µC. What is the battery voltage?

CP 32 Two electrodes connected to a 9.0 V battery are charged to ±45 nC. What is the capacitance of the electrodes?

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CP 35 A science-fair radio uses a homemade capacitor made of two 35 cm × 35 cm sheets of aluminum foil separated by a 0.25 mm-thick sheet of paper. What is the capacitance? (Note that the dielectric constant for paper is 3.0.)

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CP 67 A proton is released from rest at the positive plate of a parallel-plate capacitor. It crosses the capacitor and reaches the negative plate with a speed of 50,000 m/s. What will be the proton's final speed if the experiment is repeated with double the amount of charge on each capacitor plate?

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CP 80 The dielectric in a capacitor serves two purposes. It increases the capacitance, compared to an otherwise identical capacitor with an air gap, and it increases the maximum

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capacitor with an air gap, and it increases the maximum potential difference the capacitor can support. If the electric field in a material is sufficiently strong, the material will suddenly become able to conduct, creating a spark. The critical field strength, at which breakdown occurs, is 3.0 MV/m for air, but 60 MV/m for Teflon.(a) A parallel-plate capacitor consists of two square plates, 15 cm on a side, spaced 0.50 mm apart with only air between them. What is the maximum energy that can be stored by the capacitor?

(b) What is the maximum energy that can be stored if the plates are separated by a 0.50 mm thick Teflon sheet.

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CP 81 The flash unit in a camera uses a special circuit to "step up" the 3.0 V from the batteries to 300 V, which charges a capacitor. The capacitor is then discharged through a flash bulb. The discharge takes 10 µs, and the average power dissipated in the flash bulb is 105 W. What is the capacitance of the capacitor?

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Key Concepts

electric potential and electric potential energy-

equipotential surfaces and their properties-

relation between electric field and electric potential-

parallel-plate capacitors-

electric potential inside a conductor at equilibrium-

how a dielectric changes the properties of a capacitor-

energy stored in a capacitor-

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CP 21 In the figure, the electric potential at point A

is 300 V. Determine the potential at point B, which is 5.0 cm to the right of A.

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CP 23 Determine the magnitude and direction of the electric field at the dot in the figure.

CP 37 Two 2.0-cm-diameter electrodes with a 0.10-mm-thick sheet of Teflon between them are attached to a 9.0 V battery. Without disconnecting the battery, the Teflon is removed. Determine the charge, potential difference, and electric field (a) before and (b) after the Teflon is removed.

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CP 55 A 10.0 nC point charge and a +20.0 nC point charge are 15.0 cm apart on the x-axis. (a) Determine the electric potential at the point on the x-axis where the electric field is zero. (b) Determine the magnitude and direction of the electric field at the point on the x-axis, between the charges, where the electric potential is zero.

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CP 10 A proton with initial speed 800 000 m/s is brought to rest by an electric field. (a) Did the proton move into a region of higher potential or lower potential? (b) Determine the potential difference that stopped the proton. (c.) Determine the initial kinetic energy of the proton in electron volts.

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