# Potential difference and electric field relationship problems

### Electric Potential and Electric Field The minus sign came in because the electric field points from a region of .. This work is the charge times potential difference between the points a and b. We have seen that the difference in electric potential between two arbitrary Let us investigate the relationship between electric potential and the electric field. A charge accelerated by an electric field is analogous to a mass going down a hill. . The relationship between potential difference (or voltage) and electrical potential . considering energy can give us insights and facilitate problem solving.

It is no wonder that we do not ordinarily observe individual electrons with so many being present in ordinary systems. In fact, electricity had been in use for many decades before it was determined that the moving charges in many circumstances were negative.

Positive charge moving in the opposite direction of negative charge often produces identical effects; this makes it difficult to determine which is moving or whether both are moving. But on a submicroscopic scale, such energy per particle electron, proton, or ion can be of great importance.

For example, even a tiny fraction of a joule can be great enough for these particles to destroy organic molecules and harm living tissue. The particle may do its damage by direct collision, or it may create harmful X-rays, which can also inflict damage.

It is useful to have an energy unit related to submicroscopic effects. An electron is accelerated between two charged metal plates, as it might be in an old-model television tube or oscilloscope. The electron gains kinetic energy that is later converted into another form—light in the television tube, for example. A typical electron gun accelerates electrons using a potential difference between two separated metal plates. The energy of the electron in electron-volts is numerically the same as the voltage between the plates.

For example, a V potential difference produces eV electrons.

## Electric charge, field, and potential

The conceptual construct, namely two parallel plates with a hole in one, is shown in awhile a real electron gun is shown in b.

The Electron-Volt Unit On the submicroscopic scale, it is more convenient to define an energy unit called the electron-volt eVwhich is the energy given to a fundamental charge accelerated through a potential difference of 1 V. It follows that an electron accelerated through 50 V gains 50 eV. A potential difference ofV kV gives an electron an energy ofeV keVand so on.

### Electric Potential Difference

Similarly, an ion with a double positive charge accelerated through V gains eV of energy. These simple relationships between accelerating voltage and particle charges make the electron-volt a simple and convenient energy unit in such circumstances. The electron-volt is commonly employed in submicroscopic processes—chemical valence energies and molecular and nuclear binding energies are among the quantities often expressed in electron-volts.

For example, about 5 eV of energy is required to break up certain organic molecules. Nuclear decay energies are on the order of 1 MeV 1, eV per event and can thus produce significant biological damage. Conservation of Energy The total energy of a system is conserved if there is no net addition or subtraction due to work or heat transfer.

For conservative forces, such as the electrostatic force, conservation of energy states that mechanical energy is a constant. A loss of U for a charged particle becomes an increase in its K. As we have found many times before, considering energy can give us insights and facilitate problem solving. Electrical Potential Energy Converted into Kinetic Energy Calculate the final speed of a free electron accelerated from rest through a potential difference of V.

Assume that this numerical value is accurate to three significant figures. Strategy We have a system with only conservative forces. Assuming the electron is accelerated in a vacuum, and neglecting the gravitational force we will check on this assumption laterall of the electrical potential energy is converted into kinetic energy. From the discussion of electric charge and electric field, we know that electrostatic forces on small particles are generally very large compared with the gravitational force. The large final speed confirms that the gravitational force is indeed negligible here.

The large speed also indicates how easy it is to accelerate electrons with small voltages because of their very small mass.

Voltages much higher than the V in this problem are typically used in electron guns. These higher voltages produce electron speeds so great that effects from special relativity must be taken into account and will be discussed elsewhere.

That is why we consider a low voltage accurately in this example. A positron is identical to an electron except the charge is positive. Answer It would be going in the opposite direction, with no effect on the calculations as presented. Voltage and Electric Field So far, we have explored the relationship between voltage and energy. Now we want to explore the relationship between voltage and electric field. Consider the special case of a positive point charge q at the origin.

This will be explored further in the next section. Examining this situation will tell us what voltage is needed to produce a certain electric field strength.

It will also reveal a more fundamental relationship between electric potential and electric field. For a charge that is moved from plate A at higher potential to plate B at lower potential, a minus sign needs to be included as follows: Note that the magnitude of the electric field, a scalar quantity, is represented by E.

But, as noted earlier, arbitrary charge distributions require calculus. We therefore look at a uniform electric field as an interesting special case. In uniform E-field only: Note that this equation implies that the units for electric field are volts per meter.

We already know the units for electric field are newtons per coulomb; thus, the following relation among units is valid: The arc for calculating the potential difference between two points that are equidistant from a point charge at the origin. Above that value, the field creates enough ionization in the air to make the air a conductor. This allows a discharge or spark that reduces the field.

What, then, is the maximum voltage between two parallel conducting plates separated by 2. Strategy We are given the maximum electric field E between the plates and the distance d between them. Significance One of the implications of this result is that it takes about 75 kV to make a spark jump across a 2. This limits the voltages that can exist between conductors, perhaps on a power transmission line.

A smaller voltage can cause a spark if there are spines on the surface, since sharp points have larger field strengths than smooth surfaces.

As a result of this change in potential energy, there is also a difference in electric potential between locations A and B. By definition, the electric potential difference is the difference in electric potential V between the final and the initial location when work is done upon a charge to change its potential energy. In equation form, the electric potential difference is The standard metric unit on electric potential difference is the volt, abbreviated V and named in honor of Alessandro Volta.

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One Volt is equivalent to one Joule per Coulomb. If the electric potential difference between two locations is 1 volt, then one Coulomb of charge will gain 1 joule of potential energy when moved between those two locations. If the electric potential difference between two locations is 3 volts, then one coulomb of charge will gain 3 joules of potential energy when moved between those two locations.

And finally, if the electric potential difference between two locations is 12 volts, then one coulomb of charge will gain 12 joules of potential energy when moved between those two locations. Because electric potential difference is expressed in units of volts, it is sometimes referred to as the voltage.

Electric Potential Difference and Simple Circuits Electric circuits, as we shall see, are all about the movement of charge between varying locations and the corresponding loss and gain of energy that accompanies this movement. In the previous part of Lesson 1, the concept of electric potential was applied to a simple battery-powered electric circuit. In that discussionit was explained that work must be done on a positive test charge to move it through the cells from the negative terminal to the positive terminal.

This work would increase the potential energy of the charge and thus increase its electric potential. As the positive test charge moves through the external circuit from the positive terminal to the negative terminal, it decreases its electric potential energy and thus is at low potential by the time it returns to the negative terminal. If a 12 volt battery is used in the circuit, then every coulomb of charge is gaining 12 joules of potential energy as it moves through the battery.

And similarly, every coulomb of charge loses 12 joules of electric potential energy as it passes through the external circuit. The loss of this electric potential energy in the external circuit results in a gain in light energy, thermal energy and other forms of non-electrical energy. With a clear understanding of electric potential difference, the role of an electrochemical cell or collection of cells i.

The cells simply supply the energy to do work upon the charge to move it from the negative terminal to the positive terminal.

## Electric Potential Difference

By providing energy to the charge, the cell is capable of maintaining an electric potential difference across the two ends of the external circuit. Once the charge has reached the high potential terminal, it will naturally flow through the wires to the low potential terminal. The movement of charge through an electric circuit is analogous to the movement of water at a water park or the movement of roller coaster cars at an amusement park. In each analogy, work must be done on the water or the roller coaster cars to move it from a location of low gravitational potential to a location of high gravitational potential.

Once the water or the roller coaster cars reach high gravitational potential, they naturally move downward back to the low potential location. For a water ride or a roller coaster ride, the task of lifting the water or coaster cars to high potential requires energy.

The energy is supplied by a motor-driven water pump or a motor-driven chain. In a battery-powered electric circuit, the cells serve the role of the charge pump to supply energy to the charge to lift it from the low potential position through the cell to the high potential position.

It is often convenient to speak of an electric circuit such as the simple circuit discussed here as having two parts - an internal circuit and an external circuit. The internal circuit is the part of the circuit where energy is being supplied to the charge. For the simple battery-powered circuit that we have been referring to, the portion of the circuit containing the electrochemical cells is the internal circuit. The external circuit is the part of the circuit where charge is moving outside the cells through the wires on its path from the high potential terminal to the low potential terminal.

The movement of charge through the internal circuit requires energy since it is an uphill movement in a direction that is against the electric field.

The movement of charge through the external circuit is natural since it is a movement in the direction of the electric field. When at the positive terminal of an electrochemical cell, a positive test charge is at a high electric pressure in the same manner that water at a water park is at a high water pressure after being pumped to the top of a water slide. Being under high electric pressure, a positive test charge spontaneously and naturally moves through the external circuit to the low pressure, low potential location.