Progetto Orientamento in rete Unità 1: Newton s law of gravitation and Gravitational field Unità 2: Gravitational potential energy Unità 3: Coulomb s law and Electric field Unità 4: Magnetic field Prof.ssa Orietta Proietti 1
Newton s law of universal gravitation Prof.ssa Orietta Proietti 2
Newton s law of universal gravitation Newton's law of universal gravitation states that every point mass in the universe attracts every other point mass with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. G = 6,67*10 11 N m 2 kg 2 Prof.ssa Orietta Proietti 3
Gravitational field A gravitational field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body. The field can be determined using Newton's law of universal gravitation. F = GMm/r 2 The gravitational field g around a single particle of mass M is a vector field consisting at every point of a vector pointing directly towards the particle. The magnitude of the field at every point is calculated applying the law F = ma a g = GM/r 2 = 9,80 m/s 2 = g (M,r) Earth s mass and average radius and represents the force per unit mass on any object at that point in space. Because the force field is conservative, there is a scalar potential energy per unit mass at each point in space associated with the force fields; this is called gravitational potential. Prof.ssa Orietta Proietti 4
Lines of force U=0 (distance ) Earth s surface g = F/m = GM/r 2 Prof.ssa Orietta Proietti 5
Gravitational potential energy Prof.ssa Orietta Proietti 6
Gravitational potential energy for a 2-body system, as a function of their distance, is given by: U = - G (m 1 m 2 )/r with U=0 for r The value of U is negative U U 1 r 1 0 r 2 U 2 r W = - U U and r are inversely proportional, therefore their relation is represented by a branch of an hyperbola. Prof.ssa Orietta Proietti 7
Near the surface of the Earth, we assume that the acceleration due to gravity is a constant. In this case, a simple expression for gravitational potential energy can be derived using the equation for work W = Ph The potential energy of the tile is proportional to: P: weight W = P h = mgh = U P =mg h U U 2 U 1 U=0 h 1 h 2 h Prof.ssa Orietta Proietti 8
Elastic potential energy U 1 =1/2 ks 2 1 U 2 =1/2 ks 2 2 L = 1/2 ks 12-1/2 ks 2 2 =U 1 -U 2 = - U L>0 U < 0 L<0 U > 0 F s 1 U=0 F s 2 U=0 0 s 1 F Prof.ssa Orietta Proietti 9
The work, and then the potential energy, coincides with the area of the planar region of the graphics (F-x) mg F F ks mgh h F x 1/2 ks 2 s x G (m 1 m 2 )/r r x Prof.ssa Orietta Proietti 10
In all cases considered worth the conservation of energy E Graphics E c U Freely falling body 0 h 4 h 3 h 2 h 1 E r 4 r 3 r 2 r 1 U r Gravitational field E c Prof.ssa Orietta Proietti 11
Coulomb s law and Electric field Prof.ssa Orietta Proietti 12
Coulomb s law The electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of charges and inversely proportional to the square of distance between them. The magnitude as well as the direction of electrostatic force can be expressed by using Coulomb's law by vector equation Where is the force exerted by q1 on q2 and is the unit vector along the line joining the two charges from q1 to q2. ( r ) called DIELECTRIC CONSTANT, has different values for different dielectric materials. In empty space K, constant of proportionality, is equal to ¼ 0 where 0 = 8.85 x 10-12 C 2 /Nm 2 is permittivity of empty space. Thus in S.I. system numerical, K = 8.98755 10 9 Nm 2 C -2. Prof.ssa Orietta Proietti 13
Electric field Space or region surrounding an electric charge or a charged body within which another charge experiences some electrostatic force of attraction or repulsion. Electric intensity at a point is defined as the force experienced per unit positive charge at a point placed in the electric field. E= F/q = Q 4 r 2 [E]=[ M*L*I -1 *T -3 ] SI unit measurement: N/C or V/m Prof.ssa Orietta Proietti 14
ELECTRIC LINES OF FORCE An electric line of force is an imaginary continuous line or curve drawn in an electric field such that tangent to it at any point gives the direction of the electric force at that point.the direction of a line of force is the direction along which a small free positive charge will move along the line. It is always directed from positive charge to negative charge. CHARACTERISTICS 1. Lines of force originate from a positive charge and terminate to a negative charge. 2. The tangent to the line of force indicates the direction of the electric field and electric force. 3. Electric lines of force are always normal to the surface of charged body. 4. Electric lines of force contract longitudinally. They and expand laterally. 5. Two electric lines of force cannot intersect each other. 6. Two electric lines of force proceeding in the same direction repel each other. 7. Two electric lines of force proceeding in the opposite direction attract each other. 8. The line of force are imaginary but the field it represents as real. 9. There are no lines of force inside the conductor. Prof.ssa Orietta Proietti 15
Electric fields are similar to gravitational fields - both involve action-at-a-distance forces. Two charges of opposite sign: the force is attractive U U=k 0 (q 1 q 2 )/r r L>0 + q 1 U<0 s U=0 per r F - q 2 Prof.ssa Orietta Proietti 16
Two charges of the same sign: the force is repulsive U=k 0 (q 1 q 2 )/r U + q 1 L>0 s + q 2 F U>0 r U = 0 per r Prof.ssa Orietta Proietti 17
Electric potential The electric potential at a point is equal to the electric potential energy of any charged particle at that location divided by the charge of the particle. Since the charge of the test particle has been divided out, the electric potential is a "property" related only to the electric field itself and not the test particle. The concept of electric potential is closely linked with potential energy. A test charge q has an electric potential energy U given by U=qV The potential energy and hence also the electric potential is only defined up to an additive constant: one must arbitrarily choose a position where the potential energy and the electric potential are zero. [V]=[MLT -2 LI -1 T -1 ]=[ML 2 T -3 I -1 ] SI unit: volt 1V= 1J/1C Prof.ssa Orietta Proietti 18
Equipotential Surface Any surface over which the potential is constant is called an equipotential surface. In other words, the potential difference between any two points on an equipotential surface is zero. It may be noted that an equipotential surface may be the surface of a material body or a surface drawn in an electric field. Some important properties of equipotential surfaces : 1.Work done in moving a charge over an equipotential surface is zero. 2.The electric field is always perpendicular to an equipotential surface. 3.The spacing between equipotential surfaces enables us to identify regions of strong and weak fields. 4.Two equipotential surfaces can never intersect. 5.If two equipotential surfaces could intersect, then at the point of intersection there would be two values of electric potential which is not possible. Prof.ssa Orietta Proietti 19
Point charge Examples Capacitor U=0 + Equipotential surfaces + + + + + _ V=0 d Line of force E = Q/S 0 = / 0 V= Ed E = F/q = k 0 Q/r 2 V= U/q = k 0 Q/r E=V/d - SI unit (V/m) Prof.ssa Orietta Proietti 20
Magnetic field Prof.ssa Orietta Proietti 21
Magnetic field Magnetic fields are produced by electric currents, which can be macroscopic currents in wires, or microscopic currents associated with electrons in atomic orbits. The standard SI unit for magnetic field is the Tesla. A smaller magnetic field unit is the Gauss (1 Tesla = 10 4 Gauss). Prof.ssa Orietta Proietti 22
Long straight conductor carrying current Solenoid B B = 0i/2 r Biot Savart s law B= 0 ni n, number of turns per unit length Prof.ssa Orietta Proietti 23
Loop of wire B= 0 i/2r F= 0 i 1 i 2 l / 2 d Prof.ssa Orietta Proietti 24