The electric potential along the x-axis is V = 100x2 V, where x is in meters. What is Ex at (a) x=0 m and (b) x=1 m?

Ch 26: Potential and Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 26: Potential and Field

Problem 15

Chapter 26, Problem 15

The electric potential along the x-axis is V = 100x² V, where x is in meters. What is E_x at (a) x=0 m and (b) x=1 m?

Verified step by step guidance

Step 1: Recall the relationship between electric potential (V) and electric field (E). The electric field is the negative gradient of the electric potential. Mathematically, this is expressed as:

E_x = -dV/dx

, where

E_x

is the electric field along the x-axis and

V

is the electric potential.

Step 2: Differentiate the given electric potential function

V = 100x^2

with respect to

x

. The derivative is:

dV/dx = 200x

Step 3: Substitute the value of

x = 0

into the expression for

E_x

. Using

E_x = -dV/dx

, calculate

E_x

x = 0

Step 4: Similarly, substitute the value of

x = 1

into the expression for

E_x

. Using

E_x = -dV/dx

, calculate

E_x

x = 1

Step 5: Interpret the results. Note that the electric field is a vector quantity, and its direction is determined by the negative gradient of the potential. Discuss how the values of

E_x

x = 0

and

x = 1

relate to the behavior of the electric field along the x-axis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 measured in volts (V) and indicates how much work would be done to move a charge from a reference point to a specific point in the field without any acceleration.

Electric Field

The electric field (E) is a vector field that represents the force experienced by a unit positive charge placed in the field. It is defined as the negative gradient of the electric potential, mathematically expressed as E = -dV/dx. This relationship shows how the electric field is related to the spatial change in electric potential.

Gradient

In physics, the gradient is a vector that represents the rate and direction of change in a scalar field. For electric potential, the gradient indicates how the potential changes with respect to position. In this context, calculating the gradient of the potential function V = 100x^2 allows us to determine the electric field at specific points along the x-axis.

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