To Plot Equipotential Curves for Bar Electrode, Disc Electrode and Ring Electrode
1. Aim
To plot equipotential curves around bar electrode, disc electrode, and ring electrode configurations in a conducting medium, and to study the potential distribution patterns.
2. Apparatus Used
- Electrolytic trough (filled with water or dilute electrolyte solution)
- Bar electrode (rectangular metal plate)
- Disc electrode (circular metal plate)
- Ring electrode (annular metal plate)
- Digital voltmeter/multimeter with probes
- DC power supply (0-12V)
- Connecting wires
- Probe stand with movable probe
- Graph paper for plotting
- Millimeter scale and ruler
- Plotting pins/markers
3. Diagram
Experimental Setup for Equipotential Curves
Bar Electrode
Disc Electrode
Ring Electrode
4. Theory
Equipotential curves are lines connecting points of equal electrical potential in an electric field. These curves help us visualize the electric field distribution around electrodes of different geometries.
When electrodes are placed in a conducting medium and a potential difference is applied, an electric field is established. The field lines are perpendicular to the equipotential curves, and the gradient of potential along these field lines represents the electric field intensity.
For each electrode geometry, the potential distribution follows specific patterns:
- Bar Electrode: Produces nearly parallel equipotential curves that are perpendicular to the electrode surface. The potential gradient is approximately uniform away from the edges.
- Disc Electrode: Creates radially symmetric equipotential curves. The potential varies inversely with distance from the center of the disc.
- Ring Electrode: Produces equipotential curves that form closed loops around the ring. The potential distribution is more complex, showing variations both inside and outside the ring.
The potential at any point in the field can be determined using Laplace's equation for electrostatics:
\[ \nabla^2 V = \frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} + \frac{\partial^2 V}{\partial z^2} = 0 \]
Where V is the electrical potential.
In a uniform conducting medium, the potential decreases with distance from the source electrode according to specific patterns determined by the electrode geometry:
- For a bar electrode in a semi-infinite medium, the potential follows approximately linear patterns perpendicular to the electrode.
- For a point source (approximating a small disc), the potential varies as \( V \propto \frac{1}{r} \), where r is the distance from the source.
- For a ring electrode, the potential distribution is more complex and requires numerical solutions.
5. Formula
The electric field intensity at any point is given by:
\[ \vec{E} = -\nabla V \]
For a two-dimensional case in Cartesian coordinates:
\[ \vec{E} = -\left(\frac{\partial V}{\partial x} \hat{x} + \frac{\partial V}{\partial y} \hat{y}\right) \]
Potential gradient between two points separated by distance \(dr\):
\[ \frac{dV}{dr} = -E \]
For specific electrode geometries:
Bar Electrode (assuming infinite length):
\[ V(r) \approx V_0 - k \cdot r \]
Where \(V_0\) is the potential at the electrode surface, \(r\) is the perpendicular distance from the electrode, and \(k\) is a constant depending on the medium conductivity.
Disc Electrode (far from the edges):
\[ V(r) \approx \frac{V_0 \cdot a}{r} \]
Where \(a\) is the radius of the disc and \(r\) is the distance from the center of the disc.
Ring Electrode:
For a ring of radius \(a\), the potential at a point on the axis at distance \(z\) from the center is:
\[ V(z) = \frac{V_0 \cdot a}{\sqrt{z^2 + a^2}} \]
6. Procedure
- Setup Preparation:
- Fill the electrolytic trough with water or a dilute electrolyte solution.
- Fix a sheet of graph paper on the base of the trough (protected with waterproof covering).
- Mark a coordinate system on the graph paper.
- Bar Electrode Experiment:
- Place the bar electrode in the trough according to the marked coordinates.
- Connect the electrode to the positive terminal of the DC power supply.
- Place a second electrode (reference electrode) at a suitable distance and connect it to the negative terminal.
- Set the DC power supply to approximately 6-10V.
- Using the movable probe connected to the voltmeter, measure the potential at various points in the trough.
- Record the coordinates and corresponding potential readings in the observation table.
- Identify points of equal potential and mark them on the graph paper.
- Connect the points of equal potential to plot the equipotential curves.
- Disc Electrode Experiment:
- Replace the bar electrode with the disc electrode.
- Repeat steps similar to the bar electrode experiment.
- Take measurements at various radial distances and angular positions from the disc center.
- Record the coordinates and potential readings for the disc electrode.
- Plot the equipotential curves for the disc electrode configuration.
- Ring Electrode Experiment:
- Replace the disc electrode with the ring electrode.
- Repeat the measurement procedure, taking readings both inside and outside the ring.
- Record the coordinates and potential readings for the ring electrode.
- Plot the equipotential curves for the ring electrode configuration.
- Analysis:
- Compare the patterns of equipotential curves for all three electrode configurations.
- Analyze the potential gradient in different regions for each configuration.
- Calculate the electric field intensity at selected points using the potential gradient.
7. Observation Table
Table 1: Bar Electrode Configuration
| Point No. | X-coordinate (cm) | Y-coordinate (cm) | Potential V (volts) | Remarks |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 |
Table 2: Disc Electrode Configuration
| Point No. | Radial Distance r (cm) | Angular Position θ (degrees) | Potential V (volts) | Remarks |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 |
Table 3: Ring Electrode Configuration
| Point No. | X-coordinate (cm) | Y-coordinate (cm) | Potential V (volts) | Position (Inside/Outside Ring) |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 |
8. Calculations
1. Calculation of Electric Field Intensity
For selected points in each configuration, calculate the electric field intensity using the formula:
\[ E = -\frac{\Delta V}{\Delta r} \]
Where:
- \(E\) = Electric field intensity (V/m)
- \(\Delta V\) = Potential difference between two adjacent points (V)
- \(\Delta r\) = Distance between the two points (m)
2. Verification of Theoretical Relationships
For Bar Electrode:
Plot a graph of potential (V) versus perpendicular distance (r) from the electrode. Verify if the relationship is linear as expected:
\[ V(r) \approx V_0 - k \cdot r \]
For Disc Electrode:
Plot a graph of potential (V) versus reciprocal of distance (1/r) from the center. Verify if the relationship is linear as expected:
\[ V \cdot r \approx V_0 \cdot a \]
\[ V \approx \frac{V_0 \cdot a}{r} \]
For Ring Electrode:
For points along the axis of the ring, plot a graph of potential (V) versus \(\frac{1}{\sqrt{z^2 + a^2}}\). Verify if the relationship is linear as expected:
\[ V \cdot \sqrt{z^2 + a^2} \approx V_0 \cdot a \]
9. Result
Based on the observations and calculations, the following results are obtained:
- The equipotential curves for the bar electrode are approximately parallel lines perpendicular to the electrode surface, with closer spacing near the electrode edges.
- The equipotential curves for the disc electrode are approximately circular around the disc center, with the potential decreasing inversely with distance from the center.
- The equipotential curves for the ring electrode form closed loops, with complex patterns both inside and outside the ring.
- The experimental values of potential distribution for each electrode configuration [agree/disagree] with the theoretical predictions within experimental error.
- The electric field intensity is highest near the electrode surfaces and edges, and decreases with distance from the electrodes.
- The relationship between potential and distance for each electrode configuration follows the expected theoretical model as verified by the graphs.
10. Precautions
- Ensure that the power supply voltage is kept low (preferably below 12V) to avoid electrolysis and heating effects in the electrolyte.
- The electrolyte concentration should be optimum – not too high to cause rapid corrosion of electrodes, and not too low to create high resistance.
- Ensure that the electrodes are properly cleaned before use to avoid contamination.
- Fix the electrodes firmly in position to avoid any movement during the experiment.
- Ensure that the probes do not disturb the electrolyte significantly while taking measurements.
- Take readings quickly to minimize polarization effects at the electrode surfaces.
- Keep the depth of immersion of electrodes constant throughout the experiment.
- Ensure that the voltmeter has high input impedance to avoid loading effects on the circuit.
- Take multiple readings at each point and use the average value to minimize random errors.
- Maintain a constant water level in the trough throughout the experiment.
- Ensure that the graph paper is properly secured and protected from getting wet.
- Switch off the power supply when changing electrode configurations.
11. Viva Voice Questions
1. What are equipotential curves and what do they represent in an electric field?
Equipotential curves are lines connecting points of equal electrical potential in an electric field. They represent the spatial distribution of electric potential and are always perpendicular to the electric field lines.
2. Why do we use different electrode geometries in this experiment?
Different electrode geometries (bar, disc, ring) are used to study how the shape and configuration of electrodes affect the electric field and potential distribution in a conducting medium. Each geometry creates distinct patterns of equipotential curves that illustrate fundamental principles of electrostatics.
3. How are electric field lines related to equipotential curves?
Electric field lines are always perpendicular to equipotential curves. This is because the electric field vector points in the direction of maximum decrease in potential, which is perpendicular to surfaces of constant potential.
4. What is Laplace's equation and how is it relevant to this experiment?
Laplace's equation (\(\nabla^2 V = 0\)) describes the behavior of electric potential in regions where there is no charge. It is relevant to this experiment because it governs how the potential varies in the electrolyte between the electrodes, allowing us to predict and understand the observed equipotential patterns.
5. Why is the potential gradient steeper near the edges of the bar electrode?
The potential gradient is steeper near the edges of the bar electrode due to the edge effect or fringing of the electric field. Charge tends to accumulate at sharp edges and corners, creating a higher field intensity and steeper potential gradient in these regions.
6. How would the equipotential curves change if we used a conducting medium with higher resistivity?
The pattern of equipotential curves would remain similar, but the potential gradient would be steeper, meaning the curves would be closer together for the same applied voltage. This is because Ohm's law dictates that a higher resistivity medium would have a larger voltage drop per unit distance.
7. What practical applications utilize the knowledge of equipotential distributions?
Knowledge of equipotential distributions is important in electrical grounding systems, cathodic protection of pipelines, electroplating processes, design of high-voltage insulators, electrical impedance tomography in medical imaging, and geophysical exploration techniques.
8. Why do we use a high input impedance voltmeter for this experiment?
A high input impedance voltmeter draws minimal current from the circuit being measured, thus avoiding the disturbance of the potential distribution in the electrolyte. If a low impedance meter were used, it would act as a significant load and alter the very field pattern we are trying to measure.
9. How would the equipotential curves change if two identical electrodes with opposite polarities were used instead of a single electrode?
With two identical electrodes of opposite polarities, the equipotential curves would form patterns similar to those of a dipole field. The equipotential surfaces would be somewhat spherical near each electrode and would form saddle-shaped surfaces in the region between the electrodes.
10. What would happen to the equipotential curves if the experiment were conducted in a non-homogeneous medium?
In a non-homogeneous medium where conductivity varies spatially, the equipotential curves would be distorted. They would be more closely spaced in regions of lower conductivity (higher resistivity) and would refract at boundaries between regions of different conductivities, similar to how light refracts at interfaces between media of different refractive indices.