OCR A Physics A Level Topic 6.2: Electric Fields PDF

OCR A Physics A level Topic 6.2: Electric Fields (Content in italics is not mentioned specifically in the course specification but is nevertheless topical, relevant and possibly examinable) www.pmt.education Definition of the Electric Field...

OCR A Physics A level Topic 6.2: Electric Fields (Content in italics is not mentioned specifically in the course specification but is nevertheless topical, relevant and possibly examinable) www.pmt.education Definition of the Electric Field An electric field is a region of space in which charged particles are subject to an electrostatic force. Static Electricity and Induced Charges These forces can be shown to exist by generating a static charge on an object, for example, rubbing a glass rod with a silk cloth. The glass rod will lose electrons to the silk cloth and so become slightly positive. This rod can then be used to induce dipole charges and attract small neutral objects, such as pieces of paper or water from a tap. When the positive rod is close to a neutral object, the electric field due to the rod will attract electrons in the object and shift their positions to be closer to the rod. This causes the object to be more negative on the side closest to the rod and more positive on the opposite side i.e. the object remains neutral overall but becomes polar. The rod and the induced charge on the object will then attract. Electric Fields of Point Charges and Charged Spheres Protons and electrons are charged particles that can be modelled as point charges i.e. their charge exists at a single point in space. Point charges have radial fields which look like the spokes of a wheel. Electric field lines point outwards from a positive charge or inwards towards a negative charge as the direction of the field represents the direction of the electrostatic force on a positive charge at that point. The strength of the field decreases with www.pmt.education the radial distance or the distance away from the charge. This is shown by the decreasing density of the field lines, the less tightly packed the field lines are the weaker the electric field is at that point and the weaker the electrostatic force will be on a given charge. Spherically charged spheres, e.g. an ion or the metal sphere used in a Van de Graaff generator, can also be modelled as point charges as outside the sphere the electric field is radial and decreases in strength with distance away from the surface of the sphere. Electric Field Lines The diagrams above show the electric field lines around positive and negative point charges. Field lines generally point from positive to negative, are equally spaced as they exit a surface and are perpendicular to the surface. Lines that stretch from a positive charge to a negative charge show attraction whereas field lines that do not join up identify repulsion. Here are some other common geometries. Electric Field Strength The strength of an electric field, 𝐸𝐸 is defined as the force, 𝐹𝐹 applied per unit charge 𝑄𝑄 on an object and is expressed as: 𝐹𝐹 𝐸𝐸 = 𝑄𝑄 An E field has the unit NC-1 or Vm-1 (these units are interchangeable) and is a vector quantity as it has a direction, the direction a positive charge would accelerate at that point, and magnitude, the size of the force experienced at that point per unit charge. Forces between Point Charges Coulomb’s law states that any two point charges exert an electrostatic force between them that is proportional to the product of their charges and inversely proportional to the square of the distance between them. This is similar to Newton’s law of Gravitation (see topic 5.5) except that the charges are replaced by masses and that gravity is a solely attractive force hence the negative sign in the equation for the gravitational force. Coulomb’s law mathematically states that for two charges 𝑄𝑄 and 𝑞𝑞 at a distance 𝑟𝑟 apart 1 𝐹𝐹 ∝ 𝑄𝑄𝑄𝑄 and 𝐹𝐹 ∝ 𝑟𝑟 2 www.pmt.education Therefore, combining these statements and determining the constant of proportionality experimentally we reach 𝑄𝑄𝑄𝑄 1 𝐹𝐹 = 𝑘𝑘 2 where 𝑘𝑘 = = 8.99 × 109 Nm2 C−2 𝑟𝑟 4𝜋𝜋𝜀𝜀0 where 𝜀𝜀0 is the permittivity of free space (8.85 x 10-12 Fm-1). Permittivity is a constant that defines the ability of a material to become polarized and store charge. The permittivity of free space is the permittivity of a vacuum and as there is no matter in a vacuum to become polarized this is the lowest value possible. In matter generally, the permittivity can be written as a constant, the relative permittivity, 𝜀𝜀𝑟𝑟 , multiplied by the permittivity of free space. Electric Field of a Point Charge Using Coulomb’s law and the formula for electric field strength, it is possible to form an expression for the electric field of a point charge. To do this we must imagine bringing a small test charge of charge, 𝑞𝑞, towards a large point charge of charge, 𝑄𝑄, such that the effect of the test charge on the point charge is negligible i.e. the test charge does not cause the point charge to move. The electric field strength on the test charge can be written as 𝐹𝐹 𝐸𝐸 = 𝑞𝑞 Therefore, using Coulomb’s law for the force 𝑄𝑄 𝐸𝐸 = 4𝜋𝜋𝑟𝑟 2 𝜀𝜀0 www.pmt.education Gravitational and Electric Fields To explore the concept of a field it is useful to draw similarities between gravitational and electric fields. Gravitational Electrical Property that creates the Mass Charge Field Type of Field Always attractive (the + ve charge = repulsive to arrows on the field lines positive charges always point inwards - ve charge = attractive to towards the mass) positive charges Field Strength 𝐹𝐹 Force per unit positive Force per unit mass 𝑔𝑔 = 𝑚𝑚 𝐹𝐹 charge 𝐸𝐸 = 𝑞𝑞 Forces between Particles 1 1 𝐹𝐹 ∝ 𝑀𝑀𝑀𝑀, 𝐹𝐹 ∝ 𝑟𝑟2 𝐹𝐹 ∝ 𝑄𝑄𝑄𝑄, 𝐹𝐹 ∝ 𝑟𝑟2 Force and Field Strength 𝑀𝑀𝑀𝑀 1 𝑄𝑄𝑄𝑄 𝐹𝐹 = −𝐺𝐺 𝐹𝐹 = 𝑟𝑟 2 4𝜋𝜋𝜀𝜀0 𝑟𝑟 2 𝑀𝑀 1 𝑄𝑄 𝑔𝑔 = −𝐺𝐺 2 𝐸𝐸 = 𝑟𝑟 4𝜋𝜋𝜀𝜀0 𝑟𝑟 2 Shape of Field around a Point Point mass produces radial Point charge produces radial field field Uniform electric fields A uniform field is a field which has no dependence upon position i.e. the field is constant in space. A uniform electric field is produced between two parallel oppositely charged plates. A particle between such plates is subject to a constant force no matter where it resides between the plates. Work Done and Electric Potential Work done is a measure of the energy transferred to an object by a force. It is the product of the force and the distance over which the force acts in the direction of the force. 𝑊𝑊 = 𝐹𝐹𝐹𝐹 Therefore, work is done when moving a charge in an electric field. The work done per unit charge in moving a charge from one point to another in the field defines the potential difference between those two points. 𝑊𝑊 𝑉𝑉 = 𝑄𝑄 www.pmt.education The electric potential at a single point can be defined as the work done per unit charge to bring a positive charge from infinity to that point. The electric potential at infinity is often defined as 0. Hence, the potential difference between two points A and B is the difference between the electric potential at B, 𝑉𝑉𝐵𝐵 , and the electric potential at A, 𝑉𝑉𝐴𝐴 i.e. 𝑉𝑉 = 𝑉𝑉𝐵𝐵 − 𝑉𝑉𝐴𝐴. As the electric field can be defined by 𝐹𝐹 𝐸𝐸 = 𝑄𝑄 Then 𝑊𝑊 = 𝑉𝑉𝑉𝑉 = 𝐹𝐹𝐹𝐹 = 𝐸𝐸𝐸𝐸𝐸𝐸 This implies 𝑉𝑉 = 𝐸𝐸𝐸𝐸, however the electric field must be constant over the distance the charge moves therefore we cannot apply this equation to non-uniform fields. In this case of 𝑉𝑉 parallel plates, electric field strength can be defined as 𝐸𝐸 = 𝑑𝑑 where 𝑉𝑉 is the potential difference applied across the plates and 𝑑𝑑 is the distance between them. It is clear to see from this equation that the field can be defined in units of Vm-1. Motion of a Charged Particle in a Uniform Electric Field Due to the similarities of the gravitational and electric fields it is possible to model the motion of a charged particle through an electric field as similar to that of a massive particle through a gravitational field i.e. projectile motion. In Millikan’s oil drop experiment, oil droplets are charged with a small electric charge and allowed to fall under gravity in a vertical uniform electric field. The electric field is generated by two oppositely charged parallel plates to ensure that it is uniform. As these drops fall, the voltage between the plates can be increased and to produce a larger electric field in the gap. This produces an upward acting force which can be increased to balance with the force of gravity. When the oil drops hover in midair, we can deduce that the weight force downwards must be equal to the force of the electric field upwards as there is no acceleration. 𝑉𝑉𝑉𝑉 𝑚𝑚𝑚𝑚 = 𝐸𝐸𝐸𝐸 = 𝑑𝑑 The mass and voltage between the plates are known and the charge on the drop can be quantized as 𝑄𝑄 = −𝑛𝑛𝑛𝑛 i.e. that the charge is a whole number, 𝑛𝑛, of electrons gained by the oil drop. Therefore, a specific droplet of charge −𝑛𝑛𝑛𝑛 will be stopped by a voltage, 𝑉𝑉 , given by this equation 𝑚𝑚𝑚𝑚𝑚𝑚 𝑉𝑉 = 𝑛𝑛𝑛𝑛 Therefore, only certain values for the voltage will be measured as only small negative integer charges are produced on the droplets. By repeating the experiment for many www.pmt.education droplets, Millikan confirmed that the charges were all small integer multiples of a certain base value: 𝑒𝑒 = 1.6022 × 10−19 𝐶𝐶 Thus, Millikan’s oil drop experiment allowed for an accurate determination of the elementary charge. Work done on Point Charges Work is done when a charge, 𝑞𝑞, is brought in from infinity to a distance 𝑟𝑟 from a point charge, 𝑄𝑄. This work is the electrical energy of this geometry. The graph on the left shows the force needed as a function of the distance between the particle and the point charge. The work is the area underneath this graph from infinite separation to the distance between the charges, 𝑟𝑟. If the charges 𝑄𝑄 and 𝑞𝑞 are oppositely charged then the electrical energy would be negative as kinetic energy would be gained 𝑞𝑞 by the as it accelerates towards the charge 𝑄𝑄 from infinity. This means that it would take energy to separate the particles instead and the value of the potential energy would be how much energy it would take to move the particle 𝑞𝑞 from a distance 𝑟𝑟 away from the 𝑄𝑄 to infinity. www.pmt.education The electrical energy can be defined by the following. Work done by a force over a distance is given by 𝑊𝑊 = 𝐹𝐹𝐹𝐹 However Coulomb’s law states that 𝑄𝑄𝑄𝑄 𝐹𝐹 = 4𝜋𝜋𝑟𝑟 2 𝜀𝜀0 The work done and so the electrical potential energy needed to move a point charge 𝑞𝑞 from infinity to a radius 𝑟𝑟 is given by 𝑄𝑄𝑄𝑄 𝐸𝐸𝑒𝑒 = 4𝜋𝜋𝜋𝜋𝜀𝜀0 with the electrical energy defined as 𝐸𝐸𝑒𝑒 to distinguish it from the electric field. As the electric potential is defined as the electrical potential energy per unit charge 𝐸𝐸𝑒𝑒 𝑉𝑉 = 𝑞𝑞 Therefore 𝑄𝑄 𝑉𝑉 = 4𝜋𝜋𝜋𝜋𝜀𝜀0 which is shown in the graph on the right above. www.pmt.education