Electric Field Strength and Potential PDF

Physics Factsheet April 2002 www.curriculumpress.co.uk Number 33 Electric Field Strength and Potential Field theory - electric, magnetic or gravitational - is a model for explaining forces that act at a distance - that is, forces between two bodies that are not in physical contact. Electric fields a...

Physics Factsheet April 2002 www.curriculumpress.co.uk Number 33 Electric Field Strength and Potential Field theory - electric, magnetic or gravitational - is a model for explaining forces that act at a distance - that is, forces between two bodies that are not in physical contact. Electric fields are used to explain the forces that act between charges. Forces on charges Representing electric fields Fields are represented by drawing field lines (lines with arrows). The closer together the lines are the stronger the field. Field lines point in the direction a positive test charge would move. The diagrams below show electric fields for isolated positive and negative charges. There are two types of charge: positive charge (carried by particles such as protons) and negative charge (carried by particles such as electrons). The unit of charge is the coulomb (C). Experiments strongly suggest that charge comes in discrete packets - i.e. it is quantised. The smallest unit of charge is 1.6 × 10-19C - the size of the charge on an electron or proton - and all other amounts appear to come in integral multiples of this. Any object carrying a charge will experience a force when in the presence of another body also carrying a charge - like charges repel and unlike charges attract. This force is given by Coulomb's Law (Factsheet 32) - + kQ 1Q 2 r2 F is the force on either charge (N) Q1 and Q2 are the two charges (C) r is the seperation of the centres of the two charges (m) k is a constant; its value is 9 × 109 Nm2C-2 k = 1/(4πε0 ), whereε0 is the permittivity of free space. F= Notice in both cases the spacing of the field lines is not constant. As the distance from the point charge increases the field lines diverge, showing that the field becomes weaker. Types of electric field Point charges A "point charge" - i.e. a body with charge, but no size - does not really exist, but it is a useful approximation for small charged particles such as electrons. A attractive force is negative (Q1 and Q2 are of opposite signs) A repulsive force is positive (Q1 and Q2 are of the same sign) The force acts upon the line joining the centres of the two particles The electric field for such a charge can be found from Coulomb's law. The force on a test charge q in the presence of a charge Q is given by: kqQ F= 2 r But from the definition of electric field, we also have: Electric fields Any charged body generates an electric field Electric fields only act on charged particles. An electric field is a useful concept to help us describe electrical forces forces that act on charges. This may seem pointless if there are just two charges - Coulomb's law tells us the size and direction of the force. But electric fields allow us to describe the effect of a whole collection of charges, by combining their electrical fields. So: The electric force on a charge in an electric field is: F = qE F = force (N) q = charge (C) E = electric field strength (NC-1) F = qE qE =kqQ r2. This gives: E =kQ r2 for a point charge Q Parallel plate capacitor A parallel plate capacitor has a uniform field between the plates, except for close to the edges (edge effects). Accordingly, the field lines are constantly spaced. + Note that since force is a vector and charge is a scalar, electric field strength must be a vector. Note the non-uniformity of the field at the edges. For the uniform part of the field: Electric field strength (NC-1) is defined as the force (N) per unit charge (C) experienced by a test charge at that point. E= V d The direction of the electric field at any point is the direction of the force experienced by a positive charge at that point. V is the potential difference between the two plates. d is the distance between plates. Note: This shows that electric field strength can also have units of Vm-1 1 Electric Field Strength and Potential Physics Factsheet www.curriculumpress.co.uk Hollow charged sphere The net field strength inside a hollow charged sphere is zero. Electric potential You will already have met the idea of potential difference - a potential difference between two points means there is a voltage drop between them. Accordingly, electrical potential is measured in volts. Outside, the field behaves as if all the charge creating it is at a point in the centre of the sphere. The potential at a point in an electric field is defined as the work done in bringing a test unit positive charge from infinity to that point. The potential of a charge at infinity is always defined to be zero. If the charge creating an electric field is positive, then since "like charges repel", work must be done to bring a test positive charge in from infinity. However, if the charge creating the field is negative, then as "unlike charges attract", the work done to bring in a test positive charge from infinity is negative - the charge would tend to be attracted in, so work would actually be required to stop it. Variation of field with distance The graphs below show how field strength varies with distance for a point charge, a capacitor and a hollow charged sphere. Negatively charged point charges and spheres would give negative versions of the same graph. The potential due to a positive charge is positive. The potential due to a negative charge is negative. Point charge The decline of E with distance follows an inverse square law.(E ∝ 12 ) r E The definition of potential in terms of work tells us that electric potential is a measure of the potential energy per unit charge. The potential energy on a charge due to a potential V is: P.E. = qV P.E. = potential energy (J) q = charge (C) V = electric potential (V) r Since energy is a scalar and charge is a scalar, potential must also be a scalar quantity. Parallel plate capacitor E is constant whatever the position between the plates E If two points have different electrical potentials, then the potential energy of a charge must change if it moves from one point to the other. For this to happen, work must be done on or by the charge. V d The work done when a charge moves through a potential difference is given by W = Q∆V W = work done (J) Q = charge (C) ∆ V = potential difference (V) d The electrical potential energy increases if: A positive charge moves to a point of higher potential A negative charge moves to a point of lower potential r Hollow charged sphere E is zero within the sphere, and declines following inverse square law outside. Note that the work done depends only on the potential of the starting and finishing point - the route that the charge follows does not matter. E This tells us that if a charged particle is taken around a closed loop in an electric field, no work is done - since the potential of the starting point is the same as the potential of the finishing point. kQ r o2 ro (= radius of sphere) Typical Exam Question a) Explain the difference between electric potential and electric field strength b) A parallel plate capacitor has 100V across plate 2mm apart. (i) Find the strength of the field between the plates. (ii) What force would be exerted on an electron if it were positioned between the plates? (Charge on an electron = 1.6×10-19C) r ! per unit charge on a test charge at a point in the field a) EFS is the force! whereas potential is the work done !per unit charge on a test charge brought from infinity !to that point in the field. b) (i) E=V/d! = 100/(2×10-3) = 50kVm-1! ! (ii) F=qE = 1.6×10-19×50×103 != 8×10-15N! Exam Hint: - The variation of field strength with distance is commonly tested. You need to be able to recognise and reproduce the above graphs. 2 Electric Field Strength and Potential Physics Factsheet www.curriculumpress.co.uk Electric potential and the electric field Potential for point charge, parallel plate capacitor, hollow sphere Electric field = − gradient of electric potential Point charge Lines connecting points of equal potential - called equipotentials - can be drawn on field line diagrams. For a point charge Q, V = Equipotentials are at right angles to field lines kQ r The graph below shows how potential and electric field strength decrease with distance for a point charge. Equipotentials are like contour lines on a map (in fact, contour lines are gravitational equipotentials - they link points with the same height, and hence the same gravitational potential energy) The diagram below shows the field lines and equipotentials for a point positive charge: field line V or E Electric field strength (∝ 12 ) r equipotential Potential (∝ 1 r ) r In fact, the equipotentials around a point charge will be spheres, centred on the point charge. This is because all points on the sphere are the same distance from the point charge - so the work required to bring a charge to any point on the sphere will be the same. Parallel plate capacitor For a parallel plate capacitor with spacing d and potential difference V0 between the plates, the potential decreases linearly between the positive and negative plates Equipotentials tell us about energy changes - the energy required to move from one equipotential to another. Field lines tell us about forces the force on an electric charge at a particular point. The graph below shows how potential and electric field strength vary with distance for a parallel plate capacitor Typical Exam Question Electrons are accelerated and then directed between two parallel plates as part of a cathode ray tube, as shown below. E or V 100V E electrion beam 10 cm V 0V x 5cm a) For the region between the plates find: (i) the electric field strength; (ii) the force on an electron. Hollow charged sphere (b) Given that the velocity of the beam is 1× ×107ms-1 find: (i) the time spent between the plates; (ii) the acceleration of the electron; (iii)the vertical displacement of the electron as it just leaves the plates. The potential inside the sphere is constant, and equal to the potential on the surface. Outside, the potential is as if all the charge creating it is at the centre of the sphere. The graph below shows how potential and electric field strength vary with distance for a point charge. ×10-31kg Mass of an electron = 9.11× ×10-19C Charge on an electron = 1.6× E or V a)i) E =V/d = 100!/10× 10-2 = 1kVm-1! (ii) F=qE = 1.6× 10-19× 1× 103!= 1.6× 10-16N! (bi) t=d/v=5× 10-2/1× 107 = 5× 10-9 s! (ii)a=F/m != 1.6× 10-16/9.11× 10-31 = 1.76× 1014ms-2 ! (iii) Using equations of motion and applying them vertically a=1.76× 1014ms-2 u=0 t=5× 10-8s s=? s=ut+½at2 != ½× 1.76× 1014ms-2× (5× 10-9)2 != 2.2mm ! as measured from its "weight" on entering the field. V E r 3