Radial Electric Fields Any 2 charges exert a force on one another, attractive if they are opposite and repulsive if they are like charges. Around any charge we can imagine an invisible field of force (similar to gravity fields), which is an area of space in which a charge experiences a force. Around a Positive Point Charge + Around a Negative Point Charge - Field lines Field lines indicate the direction of the force acting on a +ve charge placed in the field. Field strength, hence force increases as you move closer to the charge - field lines are closer together. Equipotentials

Charles Augustin COULOMB (1736 - 1806) Physicien français

Q1 Q2 Q1 Q2 ?????? Units? F F r Opposite charges attract F F r In 1875 French scientist Coulomb measured the force acting on two charged spheres and came up with a fundemental force law which states, “the electrical force F between two point charges Q1 and Q2 a distance r apart is proportional to the product of the charges and is inversely proportional to the square of the distance apart”. Q1 Q2 F F r Opposite charges attract Q1 Q2 F F r Like charges repel F  Q1Q2 r2  F = 1 . Q1Q2 4o r2 = constant of proportionality 1 4o (o is the permittivity of free space = 8.85x10-12 C2N-1m-2) = 1 4o ?????? Units? Notice that the force can be attractive or repulsive depending on the sign of the charges. If the signs of Q1 and Q2 are inserted in the Coulomb force equation then: when F is +ve repulsion occurs when F is –ve attraction occurs

Example Two charged metal spheres are placed and held on a level wooden table such that their distances are 20cm apart. Each sphere experiences a repulsive force of 18μN. One of the sphere holds a charge of -10nC, what is the charge on the other? -10nC ? F F 20cm

Calculating a Resultant Electrical Field Strength E Electrical field strength E is a vector quantity and needs to be treated as such when calculating it’s resultant value at a point when more than one charge is considered. It is easier not to include the signs of the charges, just consider the direction of E due to each charge. HINT Work through problems on the board adding field strength arrows as you go! Harder Example - Four point charges are arranged symmetrically as shown in the diagram below. Calculate the resultant field strength at the centre (marked O). = 9 x 109 Nm2C-2 1 4o 2m O -8nC +8nC +5nC +10nC

Electrical Field Strength E We define electrical field strength E as “the force acting on unit +ve charge placed at a point in the field” and is measured in NC-1. since If we make Q1 = Q (charge setting up field) and Q2 = +1 C, we get: F = 1 . Q1Q2 4o r2 E = 1 . Q 4o r2 If Q is a positive point charge If Q is negative point charge E (NC-1) E  1 + r2 0 r (m) Said to create a repulsive field 0 r (m) - E (NC-1) Said to create an attractive field (like a G-field)

Electrical Potential Ve The electrical potential Ve at a certain position in an electric field is defined as, “the work done bringing unit +ve charge from infinity to that point”. It is measured in JC-1 or volts (V) and is given by: Ve = 1 . Q 4o r where Q is the charge setting up the field (sign needs to be included) and r is the distance from the centre of Q Easy way to see where this equation comes from: Work done per unit = force per unit x distance +ve charge (Ve) +ve charge  Ve = E r Potential is essentially the electrical potential energy that a +1 C charge would have at that point in the field. E = 1 . Q 4o r2 There should be a –ve here to indicate increasing potential opposes the field direction.

Electrical Potential Ve How the potential Ve varies within the field depends on the size and sign of the charge setting up the field, but again potential is taken as zero at infinity. Occasionally to simplify problems we take the Earth’s potential as zero (‘Earthing’) – see uniform field notes. If Q is a positive point charge If Q is negative point charge Ve (JC-1) + 0 r (m) Ve r 0 r (m) - Ve (JC-1) Ve r Potential ‘Hill’ Potential ‘Well’ Gradient = = - E Gradient = = - E Ve rises as you move towards +Q since work would have to be done move +1 C inwards overcoming the repulsive forces. Hence all potentials are positive Ve decreases as you move towards -Q since the charge would do the work for us in attracting a +1 C charge. Hence all potentials are negative.

Calculating a Net Potential Ve Electrical Potential Ve is a scalar quantity and needs to be treated as such when calculating it’s net value at a point when more than one charge is considered. The signs of the charges must be included. Example - Four point charges are arranged symmetrically as shown in the diagram below. Calculate the net potential at the centre (marked O). = 9 x 109 Nm2C-2 1 4o 2m O -8nC +8nC +5nC +10nC

If they are like charges, EPE is +ve and repulsion is occurring. Electrical Potential Energy EPE As a +ve charge is moved closer to another +ve charge its EPE rises (like pushing it up a ‘potential hill’), whereas it’s EPE would decrease if it moved closer to a –ve charge (falling down a ‘potential well’). The EPE of a system of two charges Q1 and Q2 separated by a distance r and is given by: If the charges are opposite, EPE is –ve and attraction is occurring, like e- in atoms. If they are like charges, EPE is +ve and repulsion is occurring. EPE = 1 .Q1Q2 4o r Electrical Potential Difference Ve and Energy Changes When a charge is moved between two points in an electrical field the work done (energy change) is independent of the path taken and depends only on the change in potential (Ve) it experiences. Electrical pd (Ve) between two points in a field is defined as, “the work done moving +1 C between the two points.” Change in energy (work done) when a charge = EPE = qVe q is moved between the two points Notes: 1. Change () = final – initial 2. Consideration of the charges involved allows us to establish if EPE has increased or decreased.

= 9 x 109 Nm2C-2 1 4o Example Two charged metal spheres X and Y are placed and held on a level wooden table as shown below. Calculate: the EPE of the system the potential at X due to Y the change in potential of X if it is moved 5cm to the left the change in EPE of the system -10nC +6nC 20cm X Y

Using Electrical Field Strength Graphs Field Strength around a positive point charge Field Strength around a negative point charge E (NC-1) r1 r2 r (m) r1 r2 r (m) E (NC-1) In both graphs the area shaded represents the potential difference between two positions r1 and r2. Area is essentially force per unit +ve charge times distance, which is the work done per unit +ve charge or pd.