Electromagnetism Topic 11.1 Electrostatic Potential Electric Potential due to a Point Charge The electric potential at a point in an electric field is defined as being numerically equal to the work done in bringing a unit positive charge from infinity to the point. The electric potential at a point in an electric field is defined as being numerically equal to the work done in bringing a unit positive charge from infinity to the point. Electric potential is a scalar quantity and it has the volt V as its unit. Electric potential is a scalar quantity and it has the volt V as its unit. Based on this definition, the potential at infinity is zero. Based on this definition, the potential at infinity is zero. Let us take a point r metres from a charged object. The potential at this point can be calculated using the following Let us take a point r metres from a charged object. The potential at this point can be calculated using the following Electric Field Strength and Potential Suppose that the charge +q is moved a small distance by a force F from A to B so that the force can be considered constant. Suppose that the charge +q is moved a small distance by a force F from A to B so that the force can be considered constant. The work done is given by: The work done is given by: W = Fx x W = Fx x The force F and the electric field E are oppositely directed, and we know that: The force F and the electric field E are oppositely directed, and we know that: F = q x E F = q x E Therefore, the work done can be given as: Therefore, the work done can be given as: W = qE x x = qV W = qE x x = qV Therefore E = - V / x Therefore E = - V / x This is the potential gradient. This is the potential gradient. Electric Field and Potential due to a charged sphere When the sphere becomes charged, we know that the charge distributes itself evenly over the surface. When the sphere becomes charged, we know that the charge distributes itself evenly over the surface. Therefore every part of the material of the conductor is at the same potential. Therefore every part of the material of the conductor is at the same potential. As the electric potential at a point is defined as being numerically equal to the work done in bringing a unit positive charge from infinity to that point, it has a constant value in every part of the material of the conductor, As the electric potential at a point is defined as being numerically equal to the work done in bringing a unit positive charge from infinity to that point, it has a constant value in every part of the material of the conductor, Since the potential is the same at all points on the conducting surface, then V / x is zero. But E = V / x. Since the potential is the same at all points on the conducting surface, then V / x is zero. But E = V / x. Therefore, the electric field inside the conductor is zero. There is no electric field inside the conductor. Therefore, the electric field inside the conductor is zero. There is no electric field inside the conductor. Equipotentials Regions in space where the electric potential of a charge distribution has a constant value are called equipotentials. Regions in space where the electric potential of a charge distribution has a constant value are called equipotentials. The places where the potential is constant in three dimensions are called equipotential surfaces, and where they are constant in two dimensions they are called equipotential lines. The places where the potential is constant in three dimensions are called equipotential surfaces, and where they are constant in two dimensions they are called equipotential lines. They are in some ways analogous to the contour lines on topographic maps. They are in some ways analogous to the contour lines on topographic maps. Similar also to gravitational potential. Similar also to gravitational potential. In this case, the gravitational potential energy is constant as a mass moves around the contour lines because the mass remains at the same elevation above the earth's surface. In this case, the gravitational potential energy is constant as a mass moves around the contour lines because the mass remains at the same elevation above the earth's surface. The gravitational field strength acts in a direction perpendicular to a contour line. The gravitational field strength acts in a direction perpendicular to a contour line. Similarly, because the electric potential on an equipotential line has the same value, no work can be done by an electric force when a test charge moves on an equipotential. Similarly, because the electric potential on an equipotential line has the same value, no work can be done by an electric force when a test charge moves on an equipotential. Therefore, the electric field cannot have a component along an equipotential, and thus it must be everywhere perpendicular to the equipotential surface or equipotential line.} Therefore, the electric field cannot have a component along an equipotential, and thus it must be everywhere perpendicular to the equipotential surface or equipotential line.} This fact makes it easy to plot equipotentials if the lines of force or lines of electric flux of an electric field are known. This fact makes it easy to plot equipotentials if the lines of force or lines of electric flux of an electric field are known. For example, there are a series of equipotential lines between two parallel plate conductors that are perpendicular to the electric field. For example, there are a series of equipotential lines between two parallel plate conductors that are perpendicular to the electric field. There will be a series of concentric circles that map out the equipotentials around an isolated positive sphere. There will be a series of concentric circles that map out the equipotentials around an isolated positive sphere. The lines of force and some equipotential lines for an isolated positive sphere are shown in the next figures. The lines of force and some equipotential lines for an isolated positive sphere are shown in the next figures. Analogies exist between electric and gravitational fields. (a) Inverse square law of force (a) Inverse square law of force Coulomb's law is similar in form to Newton's law of universal gravitation. Coulomb's law is similar in form to Newton's law of universal gravitation. Both are inverse square laws with 1/(4) in the electric case corresponding to the gravitational constant G. Both are inverse square laws with 1/(4) in the electric case corresponding to the gravitational constant G. The main difference is that whilst electric forces can be attractive or repulsive, gravitational forces are always attractive. The main difference is that whilst electric forces can be attractive or repulsive, gravitational forces are always attractive. Two types of electric charge are known but there is only one type of gravitational mass. Two types of electric charge are known but there is only one type of gravitational mass. By comparison with electric forces, gravitational forces are extremely weak. By comparison with electric forces, gravitational forces are extremely weak. (b) Field strength (b) Field strength The field strength at a point in a gravitational field is defined as the force acting per unit mass placed at the point. The field strength at a point in a gravitational field is defined as the force acting per unit mass placed at the point. Thus if a mass m in kilograms experiences a force F in newtons at a certain point in the earth's field, the strength of the field at that point will be F/m in newtons per kilogram. Thus if a mass m in kilograms experiences a force F in newtons at a certain point in the earth's field, the strength of the field at that point will be F/m in newtons per kilogram. This is also the acceleration a the mass would have in metres per second squared if it fell freely under gravity at this point (since F = ma). This is also the acceleration a the mass would have in metres per second squared if it fell freely under gravity at this point (since F = ma). The gravitational field strength and the acceleration due to gravity at a point thus have the same value (i.e. F/m) and the same symbol, g, is used for both. At the earth's surface g = 9.8 N kg ' = 9.8 m s 2 (vertically downwards). The gravitational field strength and the acceleration due to gravity at a point thus have the same value (i.e. F/m) and the same symbol, g, is used for both. At the earth's surface g = 9.8 N kg ' = 9.8 m s 2 (vertically downwards). (c) Field lines and equipotentials (c) Field lines and equipotentials These can also be drawn to represent gravitational fields but such fields are so weak, even near massive bodies, that there is no method of plotting field lines similar to those used for electric (and magnetic) fields. These can also be drawn to represent gravitational fields but such fields are so weak, even near massive bodies, that there is no method of plotting field lines similar to those used for electric (and magnetic) fields. Field lines for the earth are directed towards its centre and the field is spherically symmetrical. Field lines for the earth are directed towards its centre and the field is spherically symmetrical. Over a small part of the earth's surface the field can be considered uniform, the lines being vertical, parallel and evenly spaced. Over a small part of the earth's surface the field can be considered uniform, the lines being vertical, parallel and evenly spaced. (d) Potential and p.d. (d) Potential and p.d. Electric potentials and pds are measured in joules per coulomb (J C 1) or volts; Electric potentials and pds are measured in joules per coulomb (J C 1) or volts; gravitational potentials and pds are measured in joules per kilogram (J kg 1). gravitational potentials and pds are measured in joules per kilogram (J kg 1). As a mass moves away from the earth the potential energy of the earth mass system increases, transfer of energy from some other source being necessary. As a mass moves away from the earth the potential energy of the earth mass system increases, transfer of energy from some other source being necessary. If infinity is taken as the zero of gravitational potential (i.e. a point well out in space where no more energy is needed for the mass to move further away from the earth) If infinity is taken as the zero of gravitational potential (i.e. a point well out in space where no more energy is needed for the mass to move further away from the earth) then the potential energy of the system will have a negative value except when the mass is at infinity. then the potential energy of the system will have a negative value except when the mass is at infinity. At every point in the earth's field the potential is therefore negative (see expression below), a fact which is characteristic of fields that exert attractive forces. At every point in the earth's field the potential is therefore negative (see expression below), a fact which is characteristic of fields that exert attractive forces.