10 | Force Fields
IB Physics Content Guide
• Opposite charges/poles attract while like charges/poles repel
• The force between charged particles demonstrates the same relationship as the force between bodies with mass
• A force field describes the force at a location per unit mass, charge, or current
• A current flowing through a conductor produces a magnetic field
• The relative directions of current, magnetic field, and electromagnetic force can be found using the right-hand rules
10.1 – Static Electricity
I can explain how objects can become charged
I can qualitatively describe the reactions between charged particles
I can describe the process of grounding a charged object
10.2 – Electrostatic and Gravitational Force
I can use Coulomb’s Law to relate electrostatic force to particle charge and separation distance
I can use the Law of Gravitation to relate gravitation force to object mass and separation distance
I can determine the units of Coulomb’s Constant and the Gravitation Constant using unit analysis
I can describe how the sign of the calculated electrostatic force indicates attraction or repulsion
I can compare and contrast electrostatic and gravitation forces
I can discuss the impact of permittivity on Coulomb’s Constant
10.3 – Force Fields
I can calculate force between objects with a net charge or mass
I can draw the vector force field for electric and gravitational fields
I can describe the role of a test charge or test mass in representing force fields
I can describe how the magnitude of a force changes with distance from an object
I can calculate field strength with proper units around a single object
I can calculate the net field strength based on two or more objects
I can determine the location where the net field strength is zero
10.4 – Magnetism
I can describe the pole conditions required for magnetic attraction and repulsion
I can explain what happens when a dipole magnet is cut into pieces
I can describe the role of magnetic domains in magnetizing and de-magnetizing a material
I can draw in magnetic field lines around a magnet with a north and south pole
I can describe the layout of the Earth’s magnetic poles
10.5 – Right Hand Rule and Electromagnets
I can use the right-hand rule to draw in a magnetic field around a current carrying wire
I can use the right-hand rule to predict the current direction through a wire with a surrounding field
I can indicate a vector that is pointing into or out of the page
I can use the right-hand rule to predict the location of the north pole in a coil of wires
I can describe some applications of electromagnets in use today
I can describe the design factors that affect the strength of an electromagnet
10.6 – Electromagnetic Force
I can use the right-hand rule to predict the force direction on a charge moving through a field
I can use the right-hand rule to predict the force direction on a current carrying wire placed a field
I can describe the general functions of electric motors and generators
I can calculate the magnetic field strength and force on a wire or moving charged particle
I can predict the trajectory of a charged particle moving through a magnetic field at different speeds
10 | Force Fields
Data Booklet Equations:
Object 1 Charge
Object 2 Charge
N m2 C-2
k = 8.99 × 109 N m2 C-2
Permittivity of Free Space
C2 N-1 m-2
ε0 = 8.85 × 10-12 C2 N-1 m-2
Universal Law of Gravitation
Data Booklet Equation:
Object 1 Mass
Object 2 Mass
N m2 kg-2
G = 6.67 N m2 kg-2
Data Booklet Equation:
Data Booklet Equation:
Right Hand Rule:
Right Hand Rule #1
Right Hand Rule #2
Right Hand Rule #3
Magnetic field around a current carrying wire
Pole orientation for a coil of wire (electromagnet, solenoid, etc.)
Electromagnetic force direction on a wire or moving particle
Data Booklet Equations:
Magnetic Field Strength
Angle to Field
Charged Particles Moving through a Magnetic Field
Magnetic Field | Out of Screen
Magnetic Field | Into Screen
10.1 Describing Fields
Magnitude: g = GM/r², in N kg^-1.
Field lines: For a point or spherical mass M, the field is radial, with the field lines towards that mass. In the case of a planet, when very close to its surface, the planet may be considered flat and the field uniform.
Definition: “The gravitati
onal potential at a point P in a gravitational field is the work done per unit mass in bringing a small point mass from infinity to point P”.
Vg = W/m = -GM/r, in J kg^-1.
Work: the work done depends only on the change of the potential, not on the path taken.
Positive work is done on the test object, increasing the gravitational potential.
Negative work is done by the test object, decreasing the gravitational potential.
Potential energy (EP)
Definition for one body: “The gravitational potential energy of one body is the work done to bring one mass from infinity to a specific point”.
Definition for two bodies: “The gravitational potential energy of two bodies is the work done in bringing
the bodies to their present position when they were infinitely apart”.
EP = -GMm/r. (negative sign implies that force is attractive and that +GMm/r must be provided to infinitely separate them).
Magnitude: E = F/q = kQ/r².
Definition: “The electrical potential at a point P is the work done per unit charge for a small positive test charge to be brought from infinity to that point”.
May be visualized as the height of a flat surface.
Potential energy (EP)
Definition: “The electrical potential energy at a point P is the work done for a small positive test charge to be brought from infinity to that point”.
Charge sign must always be taken into account.
Explanation: long oppositely charge plates.
Field is uniform in the region between the plates.
Edge effect: field becomes weaker at the edges.
Explanation: consists of those points that have the same potential, i.e. which are at the same distance from the source (referred to as zero potential), and where masses or charges move without work being done on or by then.
Field lines are cut perpendicularly by the equipotential surfaces.
10.2 Fields at Work
Graphical interpretation of gravitational field strength and potential
Going upstream in the field (against) means going to a higher potential, so gain in the potential.
Going downstream in the field (in favour) means going to a lower, so loss in the potential.
Gradient of a graph of gravitational potential against distance is the gravitational field strength. g = -∆Vg/∆r
Inside a planet
Feeling weightless for an astronaut in orbit around the Earth is a consequence of both ship and the astronaut “falling freely”, with the same acceleration towards the center of planet, so that there is no normal force.
Orbital speed (vorbit): sqrt(GM/r).
Orbital period (Torbit): sqrt[4π²r³/(GM)].
Polar orbit: for satellites close to the Earth’s surface (100 km).
Geostationary orbit: for geosynchronous satellites, whose period is equal to 24 hours.
Total energy (ET) = kinetic energy (EK) + gravitational potential energy (EP).
ET = EK + EP = 1/2mv² – GMm/r = GMm/2r – GMm/r = – GMm/2r
Graph of the kinetic, potential and total energy of a mass in circular orbit around a planet as function of distance.
Increase in the orbit: total energy increases, potential energy increases and kinetic energy decreases.
Air friction: radius decreases, causing the total energy to decrease, potential energy to decrease and kinetic energy to increase.
Launching a body from a planet’s surface cases:
If total energy is positive: object will follow a hyperbolic path and never return.
If total energy is zero: object will follow a parabolic path to infinity, where it will stop.
If total energy is negative: object will go into a circular or elliptical orbit or crash.
Escape velocity: “minimum speed of object to escape gravitational field of planet/travel to infinity, starting at the surface of a planet, without energy input”.
Graphical interpretation of electric field strength and potential
Electric field strength is the force per unit charge, and thus, the area under the graph of electrical field strength against distance is the work per unit charge, i.e. the electric potential charge.
Inside a hollow conducting charged sphere
As the sphere is a conductor, all the surplus must reside on the outside of the sphere.
Charges will move until they are as far apart as possible and in equilibrium equidistant on the surface.
Inside a sphere, the force acting on a test charge are always equal in sizer and opposite in direction, and thus, cancel out: E = 0, which is the gradient ∆Ve/∆r, which means that V is constant.
Charges moving in magnetic and electric fields
Magnetic fields: force will be at right angles to velocity and magnetic field strength.
Circular path: when the charge’s direction is perpendicular to magnetic field strength.
Magnetic force = centripetal force
Helical path: charge’s movement when direction is not perpendicular to magnetic field strength.
Electric field produced by the uniform field in parallel plates
Only vertical acceleration, no horizontal.
Combination of magnetic and electric fields opposing each other, which may generate balance of forces and the charge may move in a horizontal path.
Inverse square law behavior
Geometric explanation: influence per unit area reduces to the power of 2.
The space around an electric charge, where it exerts a force on another charge is an electric field.
Electric force, like the gravitational force acts between the bodies that are not in contact with each other. To understand these forces, we involve the concept of force field. When a mass is present somewhere, the properties of space in vicinity can be considered to be so altered in such a way that another mass brought to this region will experience a force there. The space where alteration is caused by a mass is called its Gravitational field and any other mass is thought of as interacting with the field and not directly with the mass responsible for it.
Similarly an electric charge produces an electric field around it so that it interacts with any other charges present there. One reason it is preferable not to think of two charges as exerting forces upon each other directly is that if one of them is changed in magnitude or position, the consequent change in the forces each experiences does not occur immediately but takes a definite time to be established. This delay cannot be understood on the basis of coulomb law but can be explained by assuming (using field concept) that changes in field travel with a finite speed. (≈ 3 × 108 m / sec).
Electric field can be represented by field lines or line of force.
The direction of the field at any point is taken as the direction of the force on a positive charge at the point.
Electric field intensity due to a charge q at any position () from that charge is defined as
where is the force experienced by a small positive test charge q0 due to charge q.
Its SI unit is NC–1. It is a vector quantity.
If there are more charges responsible for the field, then
where are the electric field intensities due to charges q1, q2, q3…..respectively.
ELECTRIC LINES OF FORCE
These are the imaginary lines of force and the tangent at any point on the lines of force gives the direction of the electric field at that point.
PROPERTIES OF ELECTRIC LINES OF FORCE
- The lines of force diverge out from a positive charge and converge at a negative charge. i.e. the lines of force are always directed from higher to lower potential.
- The electric lines of force contract length wise indicating unlike charges attract each other and expand laterally indicating like charges repel each other.
- The number of lines that originate from or terminate on a charge is proportional to the magnitude of charge.
- Two electric lines of force never intersect each other.
- They begin from positive charge and end on negative charge i.e., they do not make closed loop (while magnetic field lines form closed loop).
- Where the electric lines of force are
- close together, the field is strong (see fig.1)
- far apart, the field is weak (see fig.2)
- Electric lines of force generate or terminate at charges /surfaces at right angles.
ELECTRIC FIELD FOR CONTINUOUS CHARGE DISTRIBUTION
If the charge distribution is continuous, then the electric field strength at any point may be calculated by dividing the charge into infinitesimal elements. If dq is the small element of charge within the charge distribution, then the electric field at point P at a distance r from charge element dq is
Non conducting sphere (dq is small charge element)
dq = λdl (line charge density)
= σ ds (surface charge density)
= ρdv (volume charge density)
The net field strength due to entire charge distribution is given by
where the integration extends over the entire charge distribution.
Note:- Electric field intensity due to a point charge q, at a distance (r1 + r2) where r1 is the thickness of medium of dielectric constant K1 and r2 is the thickness of medium of dielectric constant K2 as shown in fig. is given by
CALCULATION OF ELECTRIC FIELD INTENSITY FOR A DISTRIBUTION OF DIRECT AND CONTINUOUS CHARGE
- Fix origin of the coordinate system where electric field intensity is to be found.
- Draw the direction of electric field intensity due to the surrounding charges considering one charge at a time.
- Resolve the electric field intensity in x and y-axis respectively and find ΣEx and ΣEy
- The resultant intensity is and where θ is the angle between and x-axis.
- To find the force acting on the charge placed at the origin, the formula F = qE is used.
Energy in unit volume of electric field is called energy density and is given by
where E = electric field and εo= permittivity of vacuum
ELECTRIC FIELD DUE TO VARIOUS CHARGE DISTRIBUTION
- Electric Field due to an isolated point charge
- A circular ring of radius R with uniformly distributed charge
When x >> R,
[The charge on ring behaves as point charge]
E is max when . Also Emax
- A circular disc of radius R with uniformly distributed charge with surface charge density σ
- An infinite sheet of uniformly distributed charges with surface charge density σ
- A finite length of charge with linear charge density
Special case :
For Infinite length of charge,
- Due to a spherical shell of uniformly distributed charges with surface charge density σ
Ein = 0 (x < R)
- Due to a solid non conducting sphere of uniformly distributed charges with charge density ρ
- Due to a solid non-conducting cylinder with linear charge density λ
Eaxis = 0, ,
In above cases,
KEEP IN MEMORY
- If the electric lines of force are parallel and equally spaced, the field is uniform.
- If E0 and E be the electric field intensity at a point due to a point charge or a charge distribution in vacuum and in a medium of dielectric constant K then
E = KE0
- If E and E’ be the electric field intensity at a point in the two media having dielectric constant K and K’ then
- The electric field intensity at a point due to a ring with uniform charge distribution doesn’t depend upon the radius of the ring if the distance between the point and the centre of the ring is much greater than the radius of the ring. The ring simply behaves as a point charge.
- The electric field intensity inside a hollow sphere is zero but has a finite value at the surface and outside it (; x being the distance of the point from the centre of the sphere).
- The electric field intensity at a point outside a hollow sphere (or spherical shell) does not depend upon the radius of the sphere. It just behaves as a point charge.
- The electric field intensity at the centre of a non-conducting solid sphere with uniform charge distribution is zero. At other points inside it, the electric field varies directly with the distance from the centre (i.e. E ∝ x; x being the distance of the point from the centre). On the surface, it is constant but varies inversely with the square of the distance from the centre (i.e.). Note that the field doesn’t depend on the radius of the sphere for a point outside it. It simply behaves as a point charge.
- The electric field intensity at a point on the axis of non-conducting solid cylinder is zero. It varies directly with the distance from the axis inside it (i.e. E ∝ x). On the surface, it is constant and varies inversely with the distance from the axis for a point outside it (i.e. ).
MOTION OF A CHARGED PARTICLE IN AN ELECTRIC FIELD
Let a charged particle of mass m and charge q be placed in a uniform electric field, then electric force on the charge particle is
∴ acceleration, (constant)
- The velocity of the charged particle at time t is,
v = u + at = at = (Particle initially at rest) or
- Distance travelled by particle is
- Kinetic energy gained by particle,
If a charged particle is entering the electric field in perpendicular direction.
Let and the particle enters the field with speed u along x-axis.
Acceleration along Y-axis,
The initial component of velocity along y-axis is zero. Hence the deflection of the particle along y-axis after time t is ;
Distance covered by particle in x-axis,
x = ut …… (ii) ( acceleration ax = 0)
Eliminating t from equation (i) & (ii),
i.e. y ∝ x2.
This shows that the path of charged particle in perpendicular field is parabola.
If the width of the region in which the electric field exists be l then
- the particle will leave the field at a distance from its original path in the direction of field, given by
- The particle will leave the region in the direction of the tangent drawn to the parabola at the point of escape.
- The velocity of the particle at the point of escape is given b
- The direction of the particle in which it leaves the field is given by
Two equal and opposite charges separated by a finite distance constitute an electric dipole. If –q and +q are charges at distance 2l apart, then dipole moment,
Its SI unit is coulomb metre.
Its direction is from –q to +q. It is a vector quantity.
The torque τ on a dipole in uniform electric field as shown in figure is given by,
So τ is maximum, when dipole is ⊥ to field & minimum (=0) when dipole is parallel or antiparallel to field.
The work done in rotating the dipole from equilibrium through an angle dθ is given by
and from θ1 → θ2,
If θ1 = 0 i.e., equilibrium position, then
Work done in rotating an electric dipole in uniform electric field from θ1 to θ2 is W = pE (cosθ1 – cosθ2)
Potential energy of an electric dipole in an electric field is,
i.e. U = –pE cosθ
where θ is the angle betweenand .
We can also write
ELECTRIC FIELD DUE TO AN ELECTRIC DIPOLE
- Along the axial line (or end-on position)
and are parallel
when x >> l
- Along equatorial line (or broadside on position)
when x >>l
When and are anti parallel then,
Eax = 2 Eeq
- At any point (from the dipole)
Electric field intensity due to a point charge varies inversely as cube of the distance and in case of quadrupole it varies inversely as the fourth power of distance from the quadrupole.
ELECTRIC FORCE BETWEEN TWO DIPOLES
The electrostatic force between two dipoles of dipole moments p1 and p2 lying at a separation r is
when dipoles are placed coaxially
when dipoles are placed perpendicular to each other.
KEEP IN MEMORY
- The dipole moment of a dipole has a direction from the negative charge to the positive charge.
- If the separation between the charges of the dipole is increased (or decreased) K-times, the dipole moment increases (or decreases) by K-times.
- The torque experienced by a dipole placed in a uniform electric field has value always lying between zero and pE, where p is the dipole moment and E, the uniform electric field. It varies directly with the separation between the charges of the dipole.
- The work done in rotating a dipole in a uniform electric field varies from zero (minimum) to 2pE (maximum). Also, it varies directly with the separation between the charges of the dipole.
- The potential energy of the dipole in a uniform electric field always lies between +pE and –pE.
- The electric field intensity at a point due to an electric dipole varies inversely with the cube of the distance of the point from its centre if the distance is much greater than the length of the dipole.
- The electric field at a point due to a small dipole in end-on position is double of its value in broad side-on position,
i.e. EEnd-on = 2EBroad side-on
- For a small dipole, the electric field tends from infinity at a point very close to the axis of the dipole to zero at a point at infinity.
- The force between two dipoles increases (or decreases) by K4-times as the distance between them decreases (or increases) by K-times.
- Time period of a dipole in uniform electric field is
where I = moment of inertia of the dipole about the axis of rotation.
MOVING CHARGES AND MAGNETISM
MAGNETIC FIELD DUE TO CURRENT CARRYING CONDUCTOR, BIOT-SAVART’S LAW
MAGNETIC FIELD DUE TO VARIOUS CURRENT CARRYING CONDUCTORS
MAGNETIC FIELD DUE TO FINITE SIZED CONDUCTOR
MAGNETIC FIELD NEAR THE END OF A FINITE SIZED CONDUCTOR
MAGNETIC FIELD DUE TO AN INFINITELY LONG CONDUCTOR
MAGNETIC FIELD NEAR THE END OF A LONG CONDUCTOR
MAGNETIC FIELD DUE TO A CURRENT CARRYING COIL
MAGNETIC FIELD INSIDE A CURRENT CARRYING SOLENOID
- Finite size solenoid
- Near the end of a finite solenoid
- In the middle of a very long solenoid, B = μ0 n I
- Near the end of a very long solenoid
- Magnetic field in the endless solenoid (toroid) is same throughout and is μ0nI.
- Magnetic field outside a solenoid or toroid is zero.
AMPERE’S CIRCUITAL LAW
- The direction of the magnetic field at a point on one side of a conductor of any shape is equal in magnitude but opposite in direction of the field at an equidistant point on the other side of the conductor.
- If the magnetic field at a point due to a conductor of any shape is Bo if it is placed in vacuum then the magnetic field at the same point in a medium of relative permeability μr is given by .
- If the distance between the point and an infinitely long conductor is decreased (or increased) by K-times then the magnetic field at the point increases (or decreases) by K-times.
- The magnetic field at the centre of a circular coil of radius smaller than other similar coil with greater radius is more than that of the latter.
- For two circular coils of radii R1 and R2 having same current and same number of turns,where B1 and B2 are the magnetic fields at their centres.
- The magnetic field at a point outside a thick straight wire carrying current is inversely proportional to the distance but magnetic field at a point inside the wire is directly proportional to the distance.
- If in a coil the current is clockwise, it acts as a South-pole. If the current is anticlockwise, it acts as North-pole.
- No magnetic field occurs at point P, Q and R due to a thin current element .
- Magnetic field intensity in a thick current carrying conductor at any point x is
- Graph of magnetic field B versus x
- Magnetic field is zero at all points inside a current carrying hollow conductor.
MAGNITUDE AND DIRECTION OF MAGNETIC FIELD DUE TO DIFFERENT CONFIGURATION OF CURRENT CARRYING CONDUCTOR
FORCE ON A CONDUCTOR
α is the angle which the conductor makes with the direction of the field.
TORQUE ON A COIL
FORCE ACTING ON A CHARGED PARTICLE MOVING IN A UNIFORM MAGNETIC FIELD
FORCE BETWEEN TWO PARALLEL CURRENTS
- No force acts on a charged particle if it enters a magnetic field in a direction parallel or antiparallel to the field.
- A finite force acts on a charged particle if it enters a uniform magnetic field in a direction with finite angle with the field.
- If two charged particles of masses m1and m2 and charges q1 and q2 are projected in a uniform magnetic field with same constant velocity in a direction perpendicular to the field then the ratio of their radii (R1: R2) is given by
- The force on a conductor carrying current in a magnetic field is directly proportional to the current, the length of conductor and the magnetic field.
- If the distance between the two parallel conductors is decreased (or increased) by k-times then the force between them increases (or decreases) k-times.
- The momentum of the charged particle moving along the direction of magnetic field does not change, since the force acting on it due to magnetic field is zero.
- Lorentz force between two charges q1 and q2 moving with velocity v1, v2 separated by distance r is given by
- If the charges move, the electric as well as magnetic fields are produced. In case the charges move with speeds comparable to the speed of light, magnetic and electric force between them would become comparable.
- A current carrying coil is in stable equilibrium if the magnetic dipole moment , is parallel to and is in unstable equilibrium when is antiparallel to .
- Magnetic moment is independent of the shape of the loop. It depends on the area of the loop.
- A straight conductor and a conductor of any shape in the same plane and between the same two end points carrying equal current in the same direction, when placed in the same magnetic field experience the same force.
- There is net repulsion between two similar charges moving parallel to each other in spite of attractive magnetic force between them. This is because of electric force of repulsion which is much more stronger than the magnetic force.
- The speed of the charged particle can only be changed by an electric force.
MOVING COIL GALVANOMETER
- Determine the sign of charge carriers inside the conductor.
- Calculate the number of charge carriers per unit volume.