---
title: "Wenner's method of earth resistivity measurement"
subject: "High Voltage Engineering"
chapter: 2
key: "semester_8/high_voltage_engineering:wenner-earth-resistivity"
source: https://www.bctengineer.com/q/semester_8/high_voltage_engineering/wenner-earth-resistivity
site: BCT Engineering
---

# Wenner's method of earth resistivity measurement

High Voltage Engineering · Chapter 2: Electric Shocks

- Asked in: 2080 Chaitra, 4 marks; 2078 Chaitra, 8 marks; 2070 Bhadra, 8 marks (asked 3 times)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/high_voltage_engineering/wenner-earth-resistivity
- Whole chapter: https://www.bctengineer.com/md/semester_8/high_voltage_engineering/chapter-2

## Question

Explain Wenner's test method for measurement of earth resistivity with diagram and derive the relation for the resistivity.

## Short answer

Wenner’s method uses four electrodes in a line at spacing a; current I is injected via outer electrodes and voltage V measured across inner electrodes to yield resistance R = V/I. For depth b << a, earth resistivity is ρ = 2π a R, eliminating contact resistance.

## Answer

Wenner's four-electrode method measures the average resistivity of soil by passing current between two outer electrodes and measuring the voltage between two inner electrodes, all equally spaced in a straight line. It removes the effect of electrode contact resistance.

#### Arrangement
Four electrodes C1, P1, P2, C2 are driven in a straight line at equal spacing $a$, to a small depth $b$ ($b \ll a$, usually $b < a/20$). A current $I$ is passed through the outer (current) electrodes C1 and C2 from an earth tester, and the voltage $V$ between the inner (potential) electrodes P1 and P2 is measured.

```text
        Earth tester (megger)
     +---I----------------------+
     |      +---V---+           |
     |      |       |           |
     C1     P1      P2          C2
  ---|------|-------|-----------|--- ground
     |<--a->|<--a-->|<----a---->|
     depth b (b << a)
```

#### Derivation
Treat each electrode as a point source of current at the surface. Current $I$ from a point spreads into the earth over hemispheres. At distance $r$ the current density and field are
$$
J = \frac{I}{2\pi r^2}, \qquad E = \rho J = \frac{\rho I}{2\pi r^2}
$$
so the potential at distance $r$ is
$$
V(r) = \int_r^\infty E\,dr = \frac{\rho I}{2\pi r}
$$

Current $+I$ enters at C1 and $-I$ leaves at C2. Using superposition:

Potential at P1 (distance $a$ from C1, $2a$ from C2):
$$
V_{P1} = \frac{\rho I}{2\pi}\left(\frac{1}{a} - \frac{1}{2a}\right) = \frac{\rho I}{4\pi a}
$$

Potential at P2 (distance $2a$ from C1, $a$ from C2):
$$
V_{P2} = \frac{\rho I}{2\pi}\left(\frac{1}{2a} - \frac{1}{a}\right) = -\frac{\rho I}{4\pi a}
$$

Voltage between P1 and P2:
$$
V = V_{P1} - V_{P2} = \frac{\rho I}{2\pi a}
$$

The tester reads $R = V/I$, so
$$
\rho = 2\pi a R
$$
with $\rho$ in $\Omega\text{-m}$, $a$ in m and $R$ in $\Omega$.

If the electrode depth $b$ is not small compared with $a$ (image of each electrode taken), the general relation is
$$
\rho = \frac{4\pi a R}{1 + \dfrac{2a}{\sqrt{a^2+4b^2}} - \dfrac{a}{\sqrt{a^2+b^2}}}
$$
which reduces to $\rho = 2\pi aR$ when $b \ll a$.

#### Procedure and notes
- Readings are taken at several spacings ($a$ = 1, 2, 5, 10 m …) and in different directions. The current spreads roughly to a depth equal to $a$, so the change of $\rho$ with $a$ shows layered soil.
- Measurements are made in dry season to get the worst (highest) value for earthing design.
- An AC or reversing-DC tester is used to avoid polarisation and stray currents.

**Example:** with $a = 5$ m and $R = 3.2\ \Omega$, $\rho = 2\pi \times 5 \times 3.2 = 100.5\ \Omega\text{-m}$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In Wenner's method, where is current injected and voltage measured?
   A. Current through inner electrodes, voltage across outer
   B. Current and voltage both on outer electrodes
   C. Current through outer electrodes C1, C2; voltage across inner electrodes P1, P2, all equally spaced
   D. Only two electrodes used
   Answer: C. Four-electrode setup removes the effect of contact resistance. (Key point: Wenner arrangement)

2. The potential at distance r from a surface point electrode carrying I is:
   A. rho*I/(4*pi*r^2)
   B. rho*I/(2*pi*r)
   C. rho*I/r
   D. 2*pi*rho*I*r
   Answer: B. Current spreads over a hemisphere of area 2*pi*r^2, integrating E gives rho*I/(2*pi*r). (Key point: Potential from point source)

3. With V = rho*I/(2*pi*a) between P1 and P2, resistivity is:
   A. rho = 4*pi*a*R
   B. rho = 2*pi*a*R
   C. rho = pi*a*R
   D. rho = R/(2*pi*a)
   Answer: B. R = V/I, so rho = 2*pi*a*R. (Key point: Derived result)

4. Potential at P1 (a from C1, 2a from C2) is:
   A. rho*I/(2*pi*a)
   B. rho*I/(4*pi*a)
   C. zero
   D. rho*I/(6*pi*a)
   Answer: B. (rho*I/2pi)(1/a - 1/2a) = rho*I/(4*pi*a); P2 gives the negative of this. (Key point: Superposition at P1)

5. Why are readings taken at several spacings a?
   A. Current penetrates to roughly depth a, so variation of rho with a shows layered soil
   B. To calibrate the tester
   C. To avoid polarisation
   D. To increase contact resistance
   Answer: A. Larger spacing samples deeper soil. (Key point: Depth and spacing)

## Related in High Voltage Engineering

- [Methods of earth resistivity measurement](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Aearth-resistivity-methods) — High Voltage Engineering, chapter 2
- [Why measure earth resistivity](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Aearth-resistivity-necessity) — High Voltage Engineering, chapter 2

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