Ch 26: Potential and Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 26: Potential and Field

Problem 6

Chapter 26, Problem 6

What are the magnitude and direction of the electric field at the dot in FIGURE EX26.6?

Verified step by step guidance

Step 1: Understand the relationship between electric field and electric potential. The electric field (E) is related to the rate of change of electric potential (V) with respect to distance (d). Mathematically, this is expressed as:

E = \frac{-}{d} d

. The negative sign indicates the direction of the electric field is toward decreasing potential.

Step 2: Identify the change in electric potential (ΔV) and the distance (Δd) between the equipotential lines. From the figure, the potential changes from -200 V to 600 V over a vertical distance of 2 cm (1 cm above and 1 cm below the dot).

Step 3: Calculate the magnitude of the electric field using the formula:

E = \frac{Δ}{V}

. Substitute ΔV = 600 V - (-200 V) = 800 V and Δd = 2 cm = 0.02 m into the formula.

Step 4: Determine the direction of the electric field. The electric field points from higher potential to lower potential. In this case, the field points downward because the potential decreases as you move downward from 600 V to -200 V.

Step 5: Combine the magnitude and direction to describe the electric field at the dot. The magnitude is calculated using the formula, and the direction is downward based on the potential gradient.

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Key Concepts

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

Electric Field

The electric field is a vector field that represents the force exerted by an electric charge on other charges in its vicinity. It is defined as the force per unit charge and has both magnitude and direction. The direction of the electric field is away from positive charges and towards negative charges, indicating the direction a positive test charge would move.

Electric Potential

Electric potential, measured in volts (V), is the amount of electric potential energy per unit charge at a point in an electric field. It indicates how much work would be done to move a charge from a reference point to the specified point without any acceleration. The difference in electric potential between two points can help determine the electric field in that region.

Voltage Gradient

The voltage gradient refers to the rate of change of electric potential with respect to distance in a given direction. It is directly related to the electric field, as the electric field can be calculated as the negative gradient of the electric potential. A steeper voltage gradient indicates a stronger electric field, which influences the force experienced by charges in that field.

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Related Practice

Textbook Question

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?

Textbook Question

What is the potential difference between A and B?

Textbook Question

FIGURE EX26.3 is a graph of E_x. What is the potential difference between x_i=1.0 m and x_f=3.0 m?

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Textbook Question

Which point in FIGURE EX26.5, A or B, has a larger electric potential?

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Textbook Question

A −2.0 V equipotential surface and a +2.0 V equipotential surface are 1.0 mm apart. What is the electric field strength at a point halfway between the two surfaces?

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Textbook Question

Determine the magnitude and direction of the electric field at points 1 and 2 in FIGURE EX26.9.

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