Question

FIGURE EX26.12 is a graph of V versus x. Draw the corresponding graph of Ex versus x.

Accepted Answer

Step 1: Understand the relationship between electric potential (V) and electric field (E). The electric field is the negative gradient of the electric potential, mathematically expressed as $$ E_x = -\frac{dV}{dx} . $$This means the electric field is the rate of change of the potential with respect to position. Step 2: Analyze the graph of $$ V $$ versus $$ x . $$Look at the slope of the graph at different points. The slope of the $$ V $$ graph corresponds to $$ \frac{dV}{dx} $$, which will determine the value of $$ E_x . If $$the slope is positive, $$ E_x $$ will be negative, and if the slope is negative, $$ E_x $$ will be positive. Step 3: Identify regions where the slope is zero, positive, or negative. For example, if the $$ V $$ graph is flat (horizontal), $$ \frac{dV}{dx} = 0 $$, and $$ E_x $$ will be zero in that region. If the $$ V $$ graph is increasing, $$ \frac{dV}{dx} \u003e 0 $$, and $$ E_x $$ will be negative. If the $$ V $$ graph is decreasing, $$ \frac{dV}{dx} \u003c 0 $$, and $$ E_x $$ will be positive. Step 4: Sketch the $$ E_x $$ versus $$ x $$ graph based on the slopes of the $$ V $$ graph. For each region of the $$ V $$ graph, translate the slope into the corresponding value of $$ E_x . $$Ensure that the sign of $$ E_x $$ matches the direction of the slope (negative for increasing $$ V $$, positive for decreasing $$ V ). $$Step 5: Label the axes of the $$ E_x $$ graph properly. The x-axis remains the same as in the $$ V $$ graph, while the y-axis now represents $$ E_x . $$Ensure the graph reflects the correct magnitude and direction of $$ E_x $$ based on the analysis of the $$ V $$ graph.

FIGURE EX26.12 is a graph of V versus x. Draw the corresponding graph of E_x versus x.

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Key Concepts

Electric Field (Ex)

Gradient

Graph Interpretation