FARADAY'S LAW OF LINES OF INDUCTION

Excerpts

• FARADAY'S LAW OF LINES OF INDUCTION

  If we consider a portion of an electrified surface as cut off from the rest by the bounding line which surrounds it, and if from every point of this bounding line we draw a line of force, producing it till it meets the surface of some other body in a point which is said to *correspond* to the point of the body from which the line was drawn, these lines will form a tubular surface, and will cut off a certain portion from the surface of the other body corresponding to the portion of the surface of the first body, and the total electrifications of the two corresponding portions are equal in numerical magnitude but opposite in kind.

• FARADAY'S LAW OF LINES OF INDUCTION

  A system of lines of force forming a tubular surface closed at the one end by a portion of the positively electrified surface and at the other by the corresponding portion of the negative surface, is called by Faraday a *Tube of Induction*, because electric induction, according to Faraday, is that condition of the dielectric by which the electrifications of the opposed surfaces are placed in that physical relation to one another, which we express by saying that their electrifications are equal and opposite.

• FARADAY'S LAW OF LINES OF INDUCTION

  The similarity which constitutes the analogy is not between the phenomena themselves, but between the relations of these phenomena.

• FARADAY'S LAW OF LINES OF INDUCTION

  Hence in this simple case the number of cells is double the number of units of energy in the system.

• FARADAY'S LAW OF LINES OF INDUCTION

  It follows from (2) that the potential at the beginning of a tube is higher than at the end of it. Hence, no tube can return into itself, for in that case the same point would have two different potentials, which is impossible.

• FARADAY'S LAW OF LINES OF INDUCTION

  The boundary between these two regions forms what is called the neutral line, the form and position of which depend on the form and position of A and B.

• FARADAY'S LAW OF LINES OF INDUCTION

  Under the action of the inductor B part of its surface, on the side next to B, will become electrified oppositely to B; but since the algebraic sum of its electrification is zero, some other part of its surface must be electrified similarly to B.

• FARADAY'S LAW OF LINES OF INDUCTION

  *If the same system is electrified in three different ways, then if the potential at any point in the third case is the sum of the potentials in the first and second cases, the electrification of any part of the system in the third case will be the sum of the electrifications of the same part in the first and second cases.*

• FARADAY'S LAW OF LINES OF INDUCTION

  If the electric field under consideration consist of a finite portion of a dielectric medium, and if at every point of the boundary of this region the potential is given, and if the distribution of electrification within the region be also given, then the potential at any point within the region can have one and only one value consistent with these conditions.

• FARADAY'S LAW OF LINES OF INDUCTION

  If in any case we can find a distribution of potential which satisfies the given conditions, then by this theorem we are assured that this distribution is the only possible solution of the problem. Hence the importance of this theorem in the theory of electricity.

• FARADAY'S LAW OF LINES OF INDUCTION

  Hence the potential at every point outside of B *may*, consistently with the conditions, be the same as before. By our theorem, therefore, the potential at every point outside B *must* be the same, when, instead of the body A, we have a conducting surface B, raised to the potential P.

• FARADAY'S LAW OF LINES OF INDUCTION

  The charge of every part of the surface of a conductor is of the same sign as its potential, unless there is another body in the field whose potential is of the same sign but numerically greater.

• FARADAY'S LAW OF LINES OF INDUCTION

  If an uninsulated conductor is placed in the same field with a charged conductor, the charge on every part of the surface of the uninsulated conductor is of the opposite sign to the charge of the charged conductor.

• FARADAY'S LAW OF LINES OF INDUCTION

  Next, let us suppose that the body, A, instead of being uninsulated is insulated, but originally without charge. Under the action of the inductor B part of its surface, on the side next to B, will become electrified oppositely to B; but since the algebraic sum of its electrification is zero, some other part of its surface must be electrified similarly to B.

• FARADAY'S LAW OF LINES OF INDUCTION

  We have seen that the electric field is divided by the equipotential surfaces into a series of shells, like the coats of an onion, the thickness of each shell at any point being inversely as the electric force at that point.

• FARADAY'S LAW OF LINES OF INDUCTION

  By an imaginary surface is meant a surface which has no physical existence, but which may be imagined to exist in space without interfering with the physical properties of the substance which occupies that space. Thus we may imagine a vertical plane dividing a man’s head longitudinally into two equal parts, and by means of this imaginary surface we may render our ideas of the form of his head more precise, though any attempt to convert this imaginary surface into a physical one would be criminal.

• FARADAY'S LAW OF LINES OF INDUCTION

  This remarkable correspondence between the number of cells into which the tubes of induction are cut by the equipotential surfaces, and the electrical energy of the system, leads us to enquire whether the electrical energy may not have its true seat in the dielectric medium which is thus cut up into cells, each cell being a portion of the medium in which half a unit of energy is stored up.

• FARADAY'S LAW OF LINES OF INDUCTION

  It is probable that the actual motion of displacement is exceedingly small, in which case we must suppose the quantity of electricity in a cubic inch of the medium to be exceedingly great. If this is really the case the actual velocity of electricity in a telegraph wire may be very small, less, say, than the hundredth of an inch in an hour, though the signals which it transmits may be propagated with great velocity.

• FARADAY'S LAW OF LINES OF INDUCTION

  To begin with a case of extreme simplicity;---a person slow at arithmetic having to find the price of 52 yards of cotton at 7 pence a yard, if he happened to remember that there are 52 weeks and a day in a year of 365 days, might at once give the answer, 364 pence, without performing the calculation. Here there is no resemblance whatever between the quantities themselves---the weeks and the yards of cotton,---the sole resemblance is between the arithmetical relations of these quantities to others in the same question.

• FARADAY'S LAW OF LINES OF INDUCTION

  We must not conclude from the partial similarity of some of the relations of the phenomena of heat and electricity that there is any real physical similarity between the causes of these phenomena. The similarity is a similarity between relations, not a similarity between the things related.

• FARADAY'S LAW OF LINES OF INDUCTION

  ‘I went into this cube,’ he says, ‘and lived in it, but though I used lighted candles, electrometers, and all other tests of electrical states, I could not find the least influence upon them, or indication of anything particular given by them, though all the time the outside of the cube was powerfully charged and large sparks and brushes were starting off from every part of its outer surface.’

Equations

• FARADAY'S LAW OF LINES OF INDUCTION

  e(p - P)\text{,}

  For a single positively electrified body inside a closed conducting vessel, the whole number of cells the tubes of induction are cut into equals the body's charge times the difference between its potential and the vessel's potential.

• FARADAY'S LAW OF LINES OF INDUCTION

  ep + EP\text{.}

  Using E = -e, the cell count e(p - P) can be written as ep + EP, which the book notes is double the electrical energy of the system.

• FARADAY'S LAW OF LINES OF INDUCTION

  E_{BA} = -E_{AB}\text{,}

  The number of tubes passing from B to A is the negative of the number passing from A to B, so the count is signed by direction.

• FARADAY'S LAW OF LINES OF INDUCTION

  P_0E_0 + P_1E_1 + P_2E_2 + P_3E_3\text{,}

  For several electrified bodies A, B, C inside a vessel, the total number of cells equals the sum of each conductor's potential times its charge, which is double the electrical energy of the system.

• FARADAY'S LAW OF LINES OF INDUCTION

  E = -e

  The charge of the enclosing vessel's inner surface is equal and opposite to the charge of the body inside it.

• FARADAY'S LAW OF LINES OF INDUCTION

  \sigma = P_A \sigma _2 - P_B \sigma _1

  The surface density at a point P on body A equals the potential of A times the surface density from the unit-potential case, minus the potential of B times the surface density from the other unit case.

• FARADAY'S LAW OF LINES OF INDUCTION

  E_A = P_A q_A - P_B q_{AB}

  The total charge of body A is its potential times its capacity coefficient, minus the potential of B times the mutual coefficient of induction.

• FARADAY'S LAW OF LINES OF INDUCTION

  E_B = P_Bq_B - P_A q_{AB}

  The total charge of body B is its potential times its own coefficient, minus the potential of A times the mutual coefficient of induction.

• FARADAY'S LAW OF LINES OF INDUCTION

  P_A = P_B \frac{q_{AB}}{q_A}

  When the insulated body A carries no charge, its potential equals the potential of B times the ratio of the mutual coefficient of induction to the capacity of A.

• FARADAY'S LAW OF LINES OF INDUCTION

  \sigma = \frac{P_B}{q_A}(q_{AB} \sigma _2 - q_A \sigma _1)

  For insulated uncharged body A, the surface density at P is the potential of B divided by the capacity of A, times the difference of the two unit-case surface densities weighted by the coefficients.

Problems

No exercises in this chapter.