The electric field strength 1.5 cm from an electric dipole, on the axis of the dipole, is 1.5×105 N/C. If the dipole is replaced by a single charge, what magnitude charge in nC will give the same field strength 1.5 cm away?

Ch 23: The Electric Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 23: The Electric Field

Problem 5c

Chapter 23, Problem 5c

The electric field strength 1.5 cm from an electric dipole, on the axis of the dipole, is 1.5×10⁵ N/C. If the dipole is replaced by a single charge, what magnitude charge in nC will give the same field strength 1.5 cm away?

Verified step by step guidance

Step 1: Understand the problem. The electric field strength at a distance of 1.5 cm from a single charge is given as 1.5×10^5 N/C. We need to determine the magnitude of the charge (in nC) that produces this field strength.

Step 2: Recall the formula for the electric field due to a point charge:

E = \frac{k}{q} / r^{2}

, where

E

is the electric field strength,

k

is Coulomb's constant (

8.99 \times 10^9

N·m²/C²),

q

is the charge, and

r

is the distance from the charge.

Step 3: Rearrange the formula to solve for the charge

q

q = \frac{E}{\times} / k

. Substitute the given values:

E = 1.5 \times 10^{5}

N/C,

r = 0.015

m (converted from cm), and

k = 8.99 \times 10^{9}

N·m²/C².

Step 4: Perform the substitution into the formula:

q = \frac{1.5}{\times} / 8.99 \times 10^{9}

. Ensure all units are consistent (meters for distance, N/C for field strength).

Step 5: Convert the result for

q

into nanocoulombs (nC) by multiplying the value in coulombs by

10^9

. This will give the magnitude of the charge in nC that produces the same field strength.

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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 is measured in newtons per coulomb (N/C). The strength of the electric field decreases with distance from the charge, following an inverse square law for point charges.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance. The electric field produced by a dipole varies with distance and direction, and on the axis of the dipole, it can be calculated using specific formulas. Understanding dipoles is crucial for analyzing how they influence the electric field in their surroundings.

Coulomb's Law

Coulomb's Law describes the force between two point charges and is fundamental in electrostatics. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is essential for calculating the electric field generated by a single charge at a given distance.

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Related Practice

Textbook Question

What are the strength and direction of the electric field at the position indicated by the dot in FIGURE EX23.3? Specify the direction as an angle cw from horizontal.

Textbook Question

The electric field strength 1.5 cm from an electric dipole, on the axis of the dipole, is 1.5×10⁵ N/C. What is the dipole moment in nC mm?

Textbook Question

A 10-cm-long thin glass rod uniformly charged to +10 nC and a 10-cm-long thin plastic rod uniformly charged to −10 nC are placed side by side, 4.0 cm apart. What are the electric field strengths E₁ to E₃ at distances 1.0 cm, 2.0 cm, and 3.0 cm from the glass rod along the line connecting the midpoints of the two rods?

Textbook Question

An electric dipole is formed from two charges, ±q, spaced 1.0 cm apart. The dipole is at the origin, oriented along the y-axis. The electric field strength at the point (x, y)=(0 cm, 10 cm) is 360 N/C. What is the charge q? Give your answer in nC.

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Textbook Question

An electret is similar to a magnet, but rather than being permanently magnetized, it has a permanent electric dipole moment. Suppose a small electret with electric dipole moment 1.0×10⁻⁷ C m is 25 cm from a small ball charged to +25 nC, with the ball on the axis of the electric dipole. What is the magnitude of the electric force on the ball?