- Electric Charges and Force
- Electric Fields
- Electrostatic Potential
- Electric Current; Resistance
- Capacitor : Storing Electrical Energy
- *RC Circuit
- Magnetic Fields
- Magnetic Field Due to a Current
- Current due to a Magnetic Field
- Maxwell's Equations

Electric and Magnetic Fields

Electric Charges and Force

- Like charges repel each other and opposite charges attract as shown in Figure 7.1.
- Electric charge is
**absolutely conserved**. Electric charge cannot be created or destroyed. The total electric charge in the universe is a constant. However, since electric charge comes with opposite signs, one can create equal number of positive and negative charges without violating charge conservation. And this is precisely what happens in the phenomenon of*pair creation*where an electron with charge and an anti-electron with charge are simultaneously created. - Electric charge is discrete, comes in units of the
irreducible charge of the electron, which is denoted by . All observed
charges in nature is of the form

(7.1)

and from eq.(3.97) we have

Note that, like all other forces, the Coulomb force is a vector, and points along the line joining two charges, which is the radial vector. The Coulomb force has to be radial due to symmetry considerations, since the only vector in the problem is the radial vector .

Electric Fields

The presence of an electrical force is the manifestation of an electric field. Since the force due to a point charge is radial, we conclude that the electric field is radial, and is given by

As shown in Figure 7.2, we represent the electric field by lines

Since we are studying the case where there in only an electric field, we have the energy per unit volume of the electric field in empty space is given by

- The conservation of charge implies that electric field lines must start from sources (positive charges) and end on sinks (negative charges), as shown in Figures 7.4 and 7.5. The electric field lines in empty space cannot be terminated, as this would imply the presence of a negative charge. The conservation of charge also implies that the electric field lines cannot cross in empty space.
- The intensity of the electric field is proportional to the amount of charge that is generating the field. Note from eq.(7.7) that, for every volume in space, the energy of the electromagnetic field is proportional to the square of the electric field. Hence, the energy of the electric field is proportional to its intensity at some point as well as to the extent it is spread out in space.
- If two positive charges are brought close to each other, they will repel each other. Hence the field's lines repel each other, as shown in Figure 7.4. Consequently, the field has a lower energy if the charges are moved further apart, and is an expression of the repulsion of like charges.
- If a positive and negative
charge are brought close to each other, we know they will attract. This phenomenon
is realized by electric field lines
emerging from the positive charge and converging on the negative
charge, as shown in Figure 7.5. Since some of the electric field lines
start and end
on the positive and negative charges, the field's energy is reduced by bringing the
charges close to each other. In other words, the force between oppositely charged
particles is
**attractive**.

(7.8) |

Electrostatic Potential

The force on a charge due
to an electric field is given by

(7.9) |

In other words, the potential energy is proportional to the electrostatic potential . Consider the two points with electrostatic potential ; the electric field is then given by

The field is the

(7.13) |

(7.14) |

(7.15) |

(7.16) | |||

(7.17) | |||

(7.18) |

The electron-volt (eV) is the appropriate unit of energy to measure energies involved in atomic, chemical and (molecular) biological processes. The energy of a single molecule of air at room temperature is about eV.

(7.19) |

(7.20) | |||

(7.21) |

Figure 7.10 shows the Coulomb potential. Note the striking difference in the potentials due to a positive and negative charge, with the positive charge giving a ``mountain'' and the negative charge yielding a ``valley''.

(7.22) |

(7.23) |

(7.24) |

(7.25) |

Ohm's law states that if we create a potential difference of in a conductor having resistance , then a current given by eq.(7.33) will flow in the conductor. The dimension of resistance is ; the SI unit of resistance is ohms and

(7.27) | |||

(7.28) |

Electrical currents can flow in vacuum as well as in a medium. The resistance measures the relative ease with which a currents flows in a medium. Let a material of length and area have resistance of ; the

(7.29) |

Worthy to note is that superconductors are materials whose properties are explicable only by quantum theory, and have zero resistance to the flow of electrical currents. Conductors and insulators allow and block the flow of currents respectively, and can be understood based on principles of classical physics. Semiconductors lie in-between conductors and insulators, and their workings are based on quantum theory. Conductors have electrons that are not bound to the nuclei, and can hence move in response to an external electric field or a potential difference. All parts of a conductor at equilibrium are at the same potential, since any potential difference would cause currents to flow. One can have many resistors, and which can be combined in series and in parallel, with the value of the resultant resistance being derivable from Ohm's law. Energy is constantly lost due to the heating of the resistance , called Ohmic heating. How much power is expended by the battery in keeping the current flowing in the circuit? In time, charge of amount flows from the high electrostatic potential to the lower potential, hence losing potential energy . The difference in the electrostatic potential is given by the voltage of the battery, namely , and hence

(7.30) |

(7.31) | |||

(7.32) | |||

(7.33) | |||

(7.34) |

Note that the power loss in a circuit is proportional to . In other words, if we reverse the flow of current, the sign of will change to , but the loss due to Ohmic heating will be the same, no matter in which direction the current flows. The loss of power of the battery due to heating is not a net loss, but rather an example of the

To obtain the total energy of the field, one has to

If the electric field has a constant value, say , in a finite cube
with volume , then, from eq.(7.43), the total energy in
the electric field is given by

(7.36) |

(7.37) | |||

(7.38) |

A capacitor is charged by connecting it in a circuit with a battery with voltage , as shown in Figure 7.13. Note the capacitor is shown by two parallel lines with a gap to indicate that there is an insulator between the two conducting plates. A insulator in a D.C. circuit breaks the closed circuit, with no current flowing in the circuit. The conducting plate connected to the (+) terminal of the battery will attain the potential by the flow of electrons to the battery, leaving a net charge on the conducting plate. And similarly for the conducting plate connected to the (-) terminal; it will have the potential and charge . The charged capacitor will therefore have a voltage difference equal to the battery, namely equal to , with charges and on the two respective conducting plates. After the (transient) process of charging is over, there is no more flow of charges and the current in the circuit is zero. The energy stored in a capacitor can be found by determining the electric field that has been created due to charging up of the capacitor. The capacitor has a potential difference of across a distance , and, for now, let air be placed between the conducting plates. Hence the electric field is constant and points from the positively to the negatively charged conductors as shown in Figure 7.12. From eq.(7.18), the constant electric field is given by

(7.39) |

(7.40) | |||

(7.41) | |||

(7.42) |

It is known from charging up the capacitor that the total energy stored is given by

(7.43) | |||

(7.44) |

Hence, from the above equations, we see that the capacitance
of a parallel plate is given by

(7.45) |

(7.46) |

One can replace the insulating dielectric in a capacitor with a semi-conductor, and create a more complex device that stores and discharges electrical energy depending on the state of the semi-conductor.

(7.47) | |||

(7.48) | |||

(7.49) | |||

(7.50) |

Again, the power of dimensional analysis! There is an inherent time scale in the problem, namely, . Suppose we have a battery of voltage , the charging of the capacitor commences at . Let the current at time be , and the charge on the capacitor be . The potential drop across the resistor, from Ohm's law, is , and the potential drop across the capacitance is . At time the total drop across the circuit elements must equal the voltage of the battery, and is given by

(7.51) |

(7.52) | |||

(7.53) |

The charge on the capacitor plates is zero when the charging starts, and at final time of the charge reaches the equilibrium value of . In other words

(7.54) |

where in obtaining last equation above we have used . Eqn.(7.71) is a typical case of an exponential dependence on time. For , that is, for time much larger than the characteristic time for a capacitor, to a very good approximation the current is zero. One can conclude that for all practical purposes, the charging is over seconds after the the charging starts. To

(7.59) | |||

(7.60) |

Similar to the discussion on the charging of a capacitor, from the equation above we conclude that the capacitor is fully discharged in seconds after the start of the discharging process.

(7.61) | |||

(7.62) |

Table 7.3 gives the strengths of the magnetic fields typically encountered in nature.

Magnetic fields are most commonly experienced in the form of permanent magnets, which have a south (S) and a north (N) pole. Lines of magnetic field emanate from the north pole and converge on the south pole. Similar to charge, like poles repel and opposite poles attract. Since in the final analysis there are no magnetic monopoles, the permanent bar magnets are the result of microscopic electric currents that generate the observed magnetic fields.

Magnetic Field Due to a Current

(7.63) | |||

(7.64) | |||

(7.65) |

We have obtained the -field that is created by a current. A magnetic field external to the current, namely , can exerts a force on a current. Suppose the particles in the current carry charge , and are traveling at a velocity of . Then the force exerted on the charged particle by , as indicated in eq.(6.37), is given by

If two currents are brought close to each other, they will exert a magnetic force due to the operation of the above equation.

Current due to a Magnetic Field

(7.67) |

We apply Faraday's law to a simple example. Consider a constant magnetic field , say due to a permanent magnet, pointing in the -direction. Suppose for simplicity that the field in the -direction, and has been shown in Figure 7.16 by crosses indicating that the field points into the paper. The field extends from towards the positive -axis as shown in Figure 7.16. Insert a rectangular loop of wire, having resistance a and width , and with a distance inside the field. We apply a force along the negative -axis to pull the loop of wire

(7.69) |

(7.70) | |||

(7.71) | |||

(7.72) |

Ignoring a minus sign, we have from eq.(7.81)

(7.73) | |||

(7.74) |

We have a circuit in which an emf is driving a current . One should note that, unlike the stationary case where a potential difference drove a current in the circuit, in the case of induced emf we have time dependent electric and magnetic fields, and hence the electric field can no longer be reduced to an electrostatic potential. Since we have a circuit with a current , we can still apply Ohm's law to obtain

(7.75) |

(7.76) | |||

(7.77) |

From energy conservation, the power expended in pulling the wire loop out of the magnetic field must exactly equal the energy appearing as Ohmic heating, and this can be proven. If devices are placed in the circuit which are driven by the current , energy conservation then requires that additional mechanical energy will be required to pull the wire out of the magnetic field. The point of the exercise is to illustrate the essential principles of an electric power generator that transforms mechanical energy into electrical energy. Using the properties of magnetic fields, we can transform mechanical work into electrical energy as embodied in the induced current . The crucial point to note is that, without any mechanical contact, the magnetic field acts as an intermediate agency in transforming mechanical energy (involved in pulling out the wire loop) into an induced emf and a consequent electrical current. This ``catalytic'' role of magnetic fields is a rich resource for the invention of new devices and technologies.

- Electric fields are generated by positively and negatively charged particles. We discussed the relation of electric field to charge in Sections 7.3 and 7.2.
- A changing magnetic field generates an induced electric field. In Section 7.9 this equation of Maxwell was called Faradays' law of induction, and we derived the emf and induced electric current from a time varying magnetic field.
- There is no magnetic monopole. This is an experimental fact that continues to hold.
- An electric current or a changing electric field gives rise to an induced magnetic field. In Section 7.8 we studied the induced magnetic field due to a current.