FIGURE EX26.8 shows a graph of V versus x in a region of space. The potential is independent of y and z. What is E_x at (a) x=−2 cm, (b) x=0 cm, and (c) x=2 cm?

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Understand the relationship between electric field (E) and electric potential (V). The electric field in the x-direction, Ex, is given by the negative gradient of the potential:

E

_x = -rac{dV}{dx}. This means we need to calculate the slope of the V versus x graph at the specified points.

For part (a), at x = -2 cm, determine the slope of the V versus x graph at this point. Identify the change in potential (ΔV) and the change in position (Δx) around x = -2 cm, and calculate the slope:

rac{dV}{dx} = rac{\(\text{ΔV}\)}{\(\text{Δx}\)}

. Then, apply the negative sign to find Ex.

For part (b), at x = 0 cm, repeat the process. Examine the graph to find the slope of the V versus x curve at x = 0 cm. Use the same formula:

rac{dV}{dx} = rac{\(\text{ΔV}\)}{\(\text{Δx}\)}

, and apply the negative sign to determine Ex.

For part (c), at x = 2 cm, again calculate the slope of the V versus x graph at this point. Use the same method as in the previous steps: find ΔV and Δx around x = 2 cm, compute the slope, and apply the negative sign to find Ex.

Summarize the results for Ex at each point (x = -2 cm, x = 0 cm, and x = 2 cm) based on the calculated slopes. Ensure that the units of Ex are consistent, typically in volts per meter (V/m).

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Key Concepts

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

Electric Potential (V)

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 how much work would be done to move a charge from a reference point to a specific point in the field without any acceleration. Understanding the relationship between electric potential and electric field is crucial for solving problems related to electric forces.

Electric Field (E)

The electric field, represented as E, is a vector field that describes the force exerted per unit charge at any point in space. It is defined as the negative gradient of the electric potential, meaning that E = -dV/dx in one dimension. This relationship allows us to determine the electric field strength at various points by analyzing the changes in electric potential along the x-axis.

Gradient of a Function

The gradient of a function is a vector that points in the direction of the greatest rate of increase of that function and whose magnitude is the rate of increase in that direction. In the context of electric potential, the gradient helps us understand how the potential changes with respect to position. For the electric potential V(x), the gradient gives us the electric field E, which is essential for determining the field at specific points along the x-axis.