Electric fields (Topic 7A)
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A field is a region in space where a force can be exerted on objects possessing certain properties (such as charge or mass) without physical contact.
Fields are used to explain how forces can act at a distance, allowing one object to exert influence on another across space.
Fields are regions where objects will experience a force at a distance. A charged object creates an electric field.
Charged objects experience a force when in an electric field.
Electric fields are one of several forms of fields within physics that give rise to a force. There are many similarities and differences between different types of fields, such as electric fields, gravitational fields and magnetic fields.
All three follow the same principles that define fields, but have differences in the objects they act upon:
- Electric fields Charged objects
- Gravitational fields Objects with mass
- Magnetic fields Charged objects in motion and objects with magnetic poles
- Electromagnetic fields Combination of electric and magnetic fields.
An electric field can be created by rubbing a glass rod with a silk cloth, for example. The friction moves electrons from the rod to the cloth, leaving the rod positively charged (due to losing electrons) and the cloth negatively charged.
An electric field surrounds the rod and can attract small pieces of paper or a thin stream of water from a tap:
- The positively charged rod attracts the electrons within the pieces of paper, causing the electrons to shift towards the closer side of the rod. This creates a net attraction between the pieces of paper and the rod.
- Water molecules are said to be polar, meaning they have a slightly positive end and a slightly negative end. This is due to the shape of the molecule and the strong attraction between the electrons and the oxygen atom. When the electric field produced by the rod is brought close by, the negative end of the water molecule aligns with the rod and is attracted towards it. This results in a thin stream of water bending towards the rod as the attraction between the rod and water molecules brings the stream closer.
The electric field strength of an electric field at a point in space is defined as the force per unit charge experienced by a positive test charge at that point. It is a measure of the intensity of the electric field at that point and informs us how much force a positive test charge would experience within that field at that point.
A positive test charge is a hypothetical charge assumed to be positive, used to measure the strength and direction of an electric field at a particular point. It is positive by convention, so that the direction of the electric field lines aligns with the direction a positive charge would move.
The equation for the electric field strength is given by:
Where:
- is the force experienced by the positive test charge,
The unit of electric field strength is the newton per coulomb
The electric field strength is a vector quantity, possessing both magnitude and direction.
By convention, the direction of an electric field at a point in space is the direction a positive charge would move due to a force if placed at that point, as shown in the figure below.
Question walkthrough
Field Strength and Direction from Force on an Electron
Calculates the electric field strength from the force experienced by a moving electron, and determines the direction of the field.
Coulomb’s law states that any two point charges exert electrostatic forces on one another that are directly proportional to the product of their charges, and inversely proportional to the square of the distance between them.
Where:
- is the electrostatic force,
- and is the charge of each respective point charge, and
- is the separation distance.
Coulomb’s law applies to point charges, but can be valid for extended objects such as spheres. Spheres must be spherically symmetric, and the distance between the centres of two spheres must be much greater than their radii: essentially modelling them as point charges.
Coulomb’s law also only applies for stationary charges. If the charges are moving, then this introduces additional magnetic forces.
The law also assumes there are no external electric fields or other forces influencing the charges. If external fields are present, the resultant force must account for these additional interactions.
The diagram below shows the direction of the electrostatic forces between two point charges that have opposite charge and like charge:
Both charges in each case experience the same force due to Newton’s third law, which states that the charges will exert equal and opposite forces on one another.
Coulomb’s law states that the force between two point charges with charges and , separated by a distance is given by the equation:
Where:
- is the electrostatic force,
- and is the charge of each respective point charge, and
- is the separation distance.
- The constant of proportionality is the Boltzmann constant.
In Coulomb’s law, the constant of proportionality that is the Boltzmann constant, may be written in terms of the permittivity of free space . This allows one to rewrite Coulomb’s law as:
The permittivity of free space is a fundamental physical constant. It characterises the ability of an electric field to form and propagate throughout a vacuum, and how the electric fields interact. A greater value for the permittivity of free space would mean a weaker electric field for the same charges and distances.
Question walkthrough
Separation of Two Repelling Point Charges
Calculates the separation between two repelling point charges given their charge magnitudes and the force between them, using Coulomb's law.
Electric field lines are used to specify the direction of the field. Point charges produce radial electric fields.
The field lines point outwards for positive charges and inwards on negative charges.
A point charge and a uniformly charged sphere both produce a radial field. The uniformly charged sphere can be modelled as a point charge at its centre.
The field lines for the uniformly charged sphere (right) are the same as those for the point charge (left) beyond the dashed sphere that represents the edge of the charged sphere.
Due to the point charge and the uniformly charged sphere producing a radial field, we see that the electric field strength decreases with distance from the point charge and the uniformly charged sphere. The image shows that the space between the field lines increases with increasing distance from the point charge and uniformly charged sphere, indicating the field strength is decreasing.
Since the space between the field lines – and therefore the field strength – is decreasing, the field of a point charge and a uniformly charged sphere is non-uniform.
A point charge or charged metal sphere produces a radial electric field. Field lines become less dense with increasing distance from the source, meaning the strength of the field decreases with distance from the charge.
The electric field strength at a distance from the centre of the sphere is equal to the electrostatic force divided by the charge. One can substitute Coulomb’s law for the force to obtain:
The electric field strength is directly proportional to the charge and is inversely proportional to the square of the distance , i.e. the strength decreases with distance.
The electric field for a charged metal sphere and a point charge decreases with distance.
The electric field strength is inversely proportional to the square of the distance from the centre. Therefore, if we plot the electric field strength against the reciprocal of the square of the distance from the centre we obtain a linear relationship:
As can be seen, the gradient is constant as it is a straight line relationship, and is proportional to the charge Therefore, one could obtain the magnitude of the charge from a plot of against
Electric fields produced by point charges have infinite range. The strength of the electric field due to a point charge follows an inverse square law with the distance from the point charge :
We see that the electric field strength is inversely proportional to the square of the distance from the point charge.
Electric fields from point charges are radial in nature: the field strength decreases radially outwards from the point charge.
Question walkthrough
Electric Field Strength from a Point Charge
Calculates the electric field strength at a given distance from a point charge using Coulomb's law.
It is important to note that the definition of electric potential at a point in space is the work done per unit charge to bring a positive test charge from infinity to that point.
If a positive test charge is far enough from a positive charge to feel practically no electric field, it can be said to be at infinity.
Work must be done to bring towards due to the electrostatic repulsion between them.
The electric potential of at point (at a distance from the charge ) equals the work done per unit charge to bring to point from infinity.
The electric potential is defined as zero at infinity. As a positive test charge approaches another positive charge, its electric potential increases due to electrostatic repulsion, as illustrated below.
Conversely, if a positive test charge is brought towards a negative charge, the electric potential decreases from zero to a negative value. Work must be done to move the positive test charge away from the negative charge due to the electrostatic attraction, as shown below.
When two parallel conducting plates have a potential difference applied across them, a uniform electric field is created in the space between them, pointing from the positive plate to the negative plate.
It is important to note that the electric field strength between two oppositely charged parallel plates is related to the potential difference, applied across them and their separation,
A uniform electric field is represented by parallel, evenly spaced electric field lines. The image below shows a positive test charge placed between two oppositely charged plates:
The positive point charge (blue circle) in the image experiences a constant force and gains kinetic energy as it travels from the positive plate to the negative plate along the electric field lines.
The equation for the electric field strength between two oppositely charged parallel plates can be expressed in terms of the potential difference, between the two plates.
A charge experiences a force when moving between two oppositely charged parallel plates. Work is done on the charge and is equal to the force and the distance the charge moves which is the plate separation.
It is important to note that since the definition of potential difference is the work done per unit charge, we can write the electric strength between two oppositely charged parallel plates as:
The units for the electric field strength are The above equation shows the units for electric field strength are also
This equation is only applicable to oppositely charged parallel plates.
When drawing the electric field lines between two oppositely charged conducting plates, ensure that the field lines:
Question walkthrough
Deriving Electric Field Strength from Potential Difference
Derive an expression for the electric field strength between charged parallel plates in terms of potential difference, combining the definitions of electric field strength, work done, and potential difference.
The electric potential at a point is defined as the work done per unit charge in bringing a positive test charge from infinity to that point and is given by:
Where:
- is the charge,
- is the distance from the charge to the point at which the potential is measured,
- is a mathematical constant that comes from the way electric fields behave around a sphere, and
- is the permittivity of free space.
It is important to note that the equation does not depend on the test charge .
The units for electric potential are or volts .
It is important to note that the electric potential difference is defined as the work done per unit charge to move a positive test charge between two points in a particle’s electric field.
The electric potential difference between the two points A and B in the diagram above is the difference between the potentials at these points, i.e.
The work done to move a charge between two points in an electric field is equal to:
Where is the electric potential difference between the two points.
For a unit positive charge, , the work done is equal to the electric potential energy:
Question walkthrough
Finding Electric Potential Near a Nucleus
Calculate the charge of a uranium nucleus and use it to find the electric potential at a given distance from its surface.
Electric field lines, also known as lines of force, can visualise electric fields. They show the nature of the field created by charges and conductors:
- Arrows show the direction of the field.
- Field lines are always at to the surface.
- A uniform field has field lines that are parallel and equally spaced, i.e. the field strength is the same at all points.
- Field lines that are closer together represent greater field strength.
When two conductors have opposite charges, the electric field lines connect and point from the positive to the negative charge.
Electric fields can have different configurations depending on the size, shape, charge, and arrangement of the objects that produce them.
Uniform field
- A uniform field has equally spaced parallel field lines
- Can be produced between two parallel charged plates
The table below highlights the differences in non-uniform electric fields:
| Feature | Opposite charges | Like charges |
|---|---|---|
| Charge configuration | Two point charges of equal magnitude | Two point charges of equal magnitude, same sign |
| Field-line pattern | Lines emerge from the positive charge and terminate on the negative charge, forming continuous curves between them | Lines emerge from (or terminate on) each charge and curve away from the other; no lines pass directly between the two charges |
| Field at the midpoint (superposition) | Contributions from each charge point in the same direction and add, giving an enhanced field directed from to | Contributions from each charge are equal in magnitude but opposite in direction; they cancel exactly, so the net field is zero |
| Uniformity | Non-uniform: magnitude and direction vary with position; strength falls with distance from each charge | Non-uniform: magnitude and direction vary with position; strength falls with distance from each charge |
| Point of zero net field between the charges | None. The two contributions reinforce everywhere along the axis between the charges | Present at the midpoint (for equal magnitudes). A neutral point |
Capacitance is a measure of an object’s ability to store charge. A capacitor is an electrical device that stores charge.
Capacitors generally consist of two parallel metal plates separated by an electrically insulating material such as air. When charged by a cell or battery in a circuit, the plates of a capacitor store equal and opposite charges.
Capacitance is defined by the equation:
Where in a:
- parallel plate capacitor is the magnitude of charge stored on each plate and is the potential difference between the plates.
- charged conductor is the charge of the conductor and is its electric potential.
The unit of capacitance is the farad
One farad is equal to the capacitance of a capacitor that stores a charge of 1 C on each plate when the potential difference across the plates is 1 V.
It is important to note that one farad is very large, so practical capacitance values are often given in microfarads nanofarads or picofarads )
Question walkthrough
Finding Voltage from Charge and Capacitance
Rearrange C = Q/V to find the potential difference across a capacitor given its capacitance and stored charge.
The potential difference across a capacitor is directly proportional to the charge it stores.
It is important to note that while a capacitor is charging, the charge stored on the plates increases linearly with the voltage across the plates; therefore, a graph of charge against voltage will be a straight line passing through the origin.
The gradient of the charge against voltage graph is equal to the capacitance of the capacitor:
The energy stored by a capacitor ( is equal to the area under the charge–voltage graph for the capacitor.
The area under the graph is a triangle, so it is given by:
For a capacitor with a voltage across its plates storing a charge we have:
It is useful to know that generally, energy is given by:
the integral of the charge with respect to the potential difference. You will not need to know this calculus for your physics exam, but it could be helpful in your understanding or for future learning.
It is important to note that the correct equation for the energy stored by a capacitor. Many students make this mistake in their exams, losing marks as a result.
Question walkthrough
Finding Energy and Capacitance from a V-Q Graph
Use the area under a voltage-charge graph to find energy stored, then find capacitance from the gradient.
The energy stored by a capacitor is given by:
Where:
- is the charge stored by the capacitor, and
- is the voltage across the capacitor.
Substituting where is the capacitance of the capacitor gives:
Substituting gives:
All three equations are equivalent and can be used to calculate the stored energy, depending on which two quantities (charge , capacitance or voltage ) are known.
Question walkthrough
Finding Voltage from Capacitor Energy
Rearrange the capacitor energy formula to find the potential difference given the stored energy and capacitance.
Capacitors are used in various devices where rapid energy release is needed because of their ability to quickly discharge stored energy. Some common applications include:
An uncharged capacitor will accumulate charge when connected to a battery or cell. To prevent dangerously high currents, capacitors will usually be charged in series with a resistor.
As electrons build up on one plate during charging, a corresponding shortage of electrons is left on the other plate.
When a capacitor is charging:
- voltage across the capacitor increases until it reaches the voltage of the battery or cell,
- the charge on the plates of the capacitor increases, and
- the current in the circuit decreases.
Discharging a capacitor occurs when a charged capacitor is connected in a closed circuit. When discharging, electrons flow back from the negative plate to the positive plate until the potential difference across the capacitor falls to zero.
When a capacitor is discharging:
- voltage across the capacitor decreases
- charge (on the plates on the capacitor) decreases
- current (in the circuit) decreases but is in the opposite direction to when charging.
In the circuit setup shown, the capacitor will charge when connected to A and discharge when connected to B.
For both charging and discharging capacitors:
- Throughout the charging/discharging process, the charge on each plate is equal and opposite.
- The magnitude of the current decreases exponentially (but in opposite directions) during charge and discharge.
- A resistor or resistive circuit is needed to prevent dangerously high currents.
For a charging capacitor, the charge stored on the plates has a logarithmic growth rate, meaning the rate of increase slows with time.
The rate of increase (gradient of the against graph) tends to zero.
For a discharging capacitor, the charge stored on the plates decreases exponentially over time.
The rate of decrease (gradient of the against graph) tends to zero.
For a charging capacitor, the magnitude of current in the circuit decreases exponentially.
The rate of decrease (gradient of the against graph) tends to zero.
For a discharging capacitor, the current flows in the opposite direction. At the moment of discharge, the current is the negative of the initial current for the charging capacitor.
The magnitude of the current in a circuit with a discharging capacitor decreases exponentially.
The rate of decrease (gradient of the against graph) tends to zero.
It is important to note that the direction may be reversed (positive for discharging a capacitor and negative for charging a capacitor) depending on how the ammeter is connected relative to the circuit.
Investigating the charging of a capacitor.
- Connect the uncharged capacitor in series with a resistor, a power supply, an ammeter, and a switch, as shown in the image.
- Place a voltmeter in parallel with the capacitor.
- Close the switch to connect the power supply and simultaneously start a timer.
- Record values of voltage and current at regular intervals.
- Stop recording once the voltage across the capacitor reaches a steady state.
- Plot graphs of voltage against time and current against time.
- Alternatively, plot graphs of against and against . The gradient of these graphs will be
It is useful to know that instead of recording values manually, the ammeter and voltmeter can be connected to a datalogger. Once connected to a computer, more accurate graphs can be plotted, with readings taken at much smaller time intervals than is possible manually.
Investigating the discharging of a capacitor.
- Connect the fully charged capacitor in series with a resistor, an ammeter, and a switch, as shown in the image.
- Place a voltmeter in parallel with the capacitor.
- Close the switch to discharge the capacitor and simultaneously start a timer.
- Record values of voltage and current at regular intervals.
- Stop recording once the voltage across the capacitor reaches zero.
- Plot graphs of voltage against time and current against time.
- Alternativity plot graphs of against and against . The gradient of these graphs will be
Again, the ammeter and voltmeter can be connected to a datalogger, allowing more accurate graphs can be plotted.
Question walkthrough
Identifying the Discharge Current Graph
Identifies which graph correctly shows how ammeter current varies with time once a fully charged capacitor begins discharging through the switch.
The time constant of a capacitor–resistor circuit represents the time it takes for the voltage or charge on a discharging capacitor (or current in the circuit) to fall to times its initial value.
The time constant reflects how quickly energy stored in the capacitor is released or replenished.
For a charging capacitor, the voltage or charge (or current in the circuit) will be equal to times its initial value after one time constant \tau has passed.
It is important to note that mathematically, the time constant of a capacitor–resistor (measured in seconds) is given by:
Where:
- is the capacitance of the capacitor (measured in Farads), and
- is the resistance of the circuit (measured in Ohms).
It is important to note that a capacitor is considered to be fully charged or discharged after five time constants have passed. At this time, the capacitor will be over 99% charged or discharged.
Question walkthrough
Time Constant of a Series Circuit
Combines series resistors and series capacitors to find total resistance and capacitance, then calculates the time constant of the resulting RC circuit.
Question walkthrough
Finding Resistance from a Charging Graph
Uses the one-time-constant charge value read from a charging capacitor's Q-t graph to calculate the resistance of the resistor in the circuit.
For a capacitor discharging across a circuit with a resistance we have:
Where:
- is the rate of flow of charge off of the capacitor,
- is the charge on the capacitor, is the resistance of the resistor, and
- is the capacitance of the capacitor.
It is important to note that as the charge decreases in a discharging capacitor, the change in charge is negative. Its magnitude equals the current in the circuit:
The voltage across the capacitor at time as it discharges is the same as the voltage across the resistor (as the sum of the potentials in a closed loop is zero), so the current in the circuit is also given by:
For the capacitor, we also have:
Combining these equations gives:
The magnitude of the voltage charge and current for a charging capacitor as a function of time are given by:
Where:
- is the voltage of the power supply, and the final voltage across the capacitor,
- is the final charge across the capacitor,
- is the initial current in the circuit,
- is the resistance of the circuit,
- and, is the capacitance of the capacitor.
For a discharging capacitor discharging in series with a resistor, we have:
Where:
- is voltage, charge, or current, and
- is the initial voltage, charge, or current.
Taking the natural logarithm of both sides gives a linear relationship:
A graph of plotted against will have a gradient of and a -intercept of
The gradients and -intercepts are the same because they stem from the same fundamental equation governing exponential growth and decay.
For a charging capacitor charging in series with a resistor we have:
Where:
- is the voltage or charge, and
- is the initial voltage or charge.
Rearranging for and taking the natural logarithm of both sides gives a linear relationship:
A graph of plotted against will have a gradient of and a -intercept of
Question walkthrough
Finding Capacitance from Discharge Time
Rearranges the exponential discharge equation to calculate the capacitance of a capacitor given the resistance and time taken to fall to one-third of its initial voltage.
Question walkthrough
Current and Time Constant from Graph
Extracts the initial current and time constant of a charging capacitor circuit from the gradient and intercept of a linearised ln(I) against t graph.
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Deriving the One Time-Constant Voltage
Uses the exponential discharge equation to show that capacitor voltage falls to approximately 37% of its initial value after one time constant, RC.
For a discharging capacitor charging in series with a resistor we have:
Where:
- is voltage, charge or current,
- and, is the initial voltage, charge or current.
The voltage–time graph for an exponentially decaying capacitor against time shows a rapid drop initially, followed by a slower decrease over time.
The exponential decay graph has the characteristic constant-ratio property: the charge (or current/voltage) decreases by the same fraction for equal time intervals.
It is important to note that this property applies to any fixed ratio of the quantity being measured over equal time intervals:
- For the ratio the corresponding time interval is the time constant .
- For the ratio the corresponding ratio is known as the half-life.
Question walkthrough
Capacitor Charge After Repeated Discharging
Applies repeated fractional decay to find a capacitor's charge 5 seconds after discharge begins, given the charge remaining after 1 second.