Engineers discover that the electric potential between two electrodes can be modeled as V(x)=V₀ln(1+x/d) , where V₀ is a constant, x is the distance from the first electrode in the direction of the second, and d is the distance between the electrodes. What is the electric field strength midway between the electrodes?

Verified step by step guidance

1. Step 1: Recall the relationship between electric potential and electric field. The electric field strength, E(x), is the negative gradient of the electric potential, V(x). Mathematically, this is expressed as: $E(x)\; =\; -dV(x)/dx$.
2. Step 2: Write down the given electric potential function: $V(x)\; =\; V₀\; ln(1\; +\; x/d)$, where V₀ is a constant, x is the distance from the first electrode, and d is the distance between the electrodes.
3. Step 3: Differentiate the potential function with respect to x to find the electric field. Using the chain rule, the derivative of $ln(1\; +\; x/d)$ is $1/(1\; +\; x/d)\; *\; (1/d)$. Thus, $E(x)\; =\; -V₀/(d(1\; +\; x/d))$.
4. Step 4: Determine the position midway between the electrodes. Since the distance between the electrodes is d, the midpoint corresponds to $x\; =\; d/2$.
5. Step 5: Substitute $x\; =\; d/2$ into the expression for the electric field: $E(d/2)\; =\; -V₀/(d(1\; +\; (d/2)/d))$. Simplify the denominator to find the electric field strength at the midpoint.

Key Concepts

Electric Potential

Electric potential, denoted as V, is the amount of electric potential energy per unit charge at a point in an electric field. It is a scalar quantity that indicates the work done to move a charge from a reference point to a specific point in the field without any acceleration. In this context, the potential is modeled as a logarithmic function of distance, which reflects how the potential changes as one moves between the electrodes.

Electric Field Strength

Electric field strength, represented as E, is defined as the force per unit charge experienced by a positive test charge placed in the field. It is a vector quantity that points in the direction of the force that a positive charge would experience. The electric field can be derived from the electric potential by taking the negative gradient of the potential function, which indicates how rapidly the potential changes with respect to position.

Gradient and Derivatives

In physics, the gradient of a scalar field, such as electric potential, is a vector that points in the direction of the greatest rate of increase of the field and whose magnitude is the rate of increase in that direction. The derivative of the potential function with respect to distance gives the electric field strength. Understanding how to compute derivatives is essential for analyzing how physical quantities change in space, particularly in the context of electric fields.