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Ch. 23 - Electric Potential
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 23 - Electric PotentialProblem 26
Chapter 22, Problem 26

Two point charges, 3.4 μC and -2.0 μC, are placed 8.0 cm apart on the x axis. At what points along the x axis are
(a) the electric field zero and
(b) the potential zero? Let V = 0 at r = ∞.

Verified step by step guidance
1
Convert the given quantities into standard SI units: the charges are q₁ = 3.4 × 10⁻⁶ C and q₂ = -2.0 × 10⁻⁶ C, and the distance between them is d = 0.08 m.
To find where the electric field is zero (part a), recall that the electric field due to a point charge is given by E = (1 / (4πε₀)) * (q / r²). The net electric field will be zero at a point where the magnitudes of the electric fields due to both charges are equal and opposite. Set up the equation: (q₁ / r₁²) = (|q₂| / r₂²), where r₁ is the distance from q₁ to the point, and r₂ is the distance from q₂ to the point. Use the relationship r₁ + r₂ = d to express r₂ in terms of r₁ and solve for r₁.
To find where the potential is zero (part b), recall that the electric potential due to a point charge is V = (1 / (4πε₀)) * (q / r). The net potential will be zero at a point where the algebraic sum of the potentials due to both charges is zero. Set up the equation: (q₁ / r₁) + (q₂ / r₂) = 0. Again, use the relationship r₁ + r₂ = d to express r₂ in terms of r₁ and solve for r₁.
For both parts, solve the resulting equations algebraically to find the distances r₁ and r₂. Note that for part (a), the point where the electric field is zero must lie between the charges if they are of opposite signs, while for part (b), the point where the potential is zero can lie either between the charges or outside the region between them.
Verify the solutions by substituting the calculated distances back into the respective equations for electric field and potential to ensure that the conditions (E = 0 and V = 0) are satisfied.

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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 directed away from positive charges and towards negative charges. The electric field can be calculated using Coulomb's law, which describes the interaction between point charges.
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Electric Potential

Electric potential, or voltage, is a scalar quantity that represents the 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 (usually at infinity) to a specific point in the field. The potential is zero at infinity, and it can be calculated by integrating the electric field along a path.
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Superposition Principle

The superposition principle states that the total electric field or potential at a point due to multiple charges is the vector sum of the fields or potentials due to each charge individually. This principle allows us to analyze complex charge configurations by considering the contributions from each charge separately, making it essential for solving problems involving multiple point charges.
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Related Practice
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Textbook Question

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What minimum radius must a large conducting sphere (of an electrostatic generating machine) have if it is to be at 45,000 V without discharge into the air? How much charge will it carry?

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A very long conducting cylinder (length ℓ) of radius R₀ (R₀ ≪ ℓ) carries a uniform surface charge density σ (C/m²). The cylinder is at an electric potential V₀. Determine the potential, at points far from the end, at a distance R from the center of the cylinder for

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Textbook Question

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