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88. Due cariche puntiformi rispettivamente di+3,40μC\mathrm{di}+3,40 \mu \mathrm{C} $\...
May 5, 2024
88 Answer
The electric potential at the midpoint is 0 V.
Solution
a
Use the formula for electric potential due to a point charge: V=kQrV = \frac{kQ}{r}, where kk is Coulomb's constant, QQ is the charge, and rr is the distance from the charge
b
Calculate the potential at the midpoint due to each charge separately and then add them up. Since the charges are equal and opposite, and the midpoint is equidistant from both, the potentials due to each charge will cancel each other out
Key Concept
Superposition principle in electric potential
Explanation
The electric potential at a point is the algebraic sum of the potentials due to each charge. When two equal but opposite charges are equidistant from a point, their potentials cancel each other out, resulting in zero net potential at that point.
89 Answer
The value of the charge is 5.40×10−5 C5.40 \times 10^{-5} \, \mathrm{C}.
Solution
a
Use the work-energy theorem: W=qΔVW = q\Delta V, where WW is the work done, qq is the charge, and ΔV\Delta V is the potential difference
b
Rearrange the formula to solve for the charge: q=WΔVq = \frac{W}{\Delta V}
c
Substitute the given values: q=2.70×10−3 J50.0 Vq = \frac{2.70 \times 10^{-3} \, \mathrm{J}}{50.0 \, \mathrm{V}}
Key Concept
Relationship between work, charge, and potential difference
Explanation
The charge can be calculated by dividing the work done by the electric force by the potential difference between two points.
90 Answer
The potential difference between the plates of the capacitor is 1.1×103 V1.1 \times 10^{3} \, \mathrm{V}.
Solution
a
Use the formula for energy stored in a capacitor: U=12CV2U = \frac{1}{2}CV^2, where UU is the stored energy, CC is the capacitance, and VV is the potential difference
b
Rearrange the formula to solve for VV: V=2UCV = \sqrt{\frac{2U}{C}}
c
Substitute the given values: V=2×73 J120×10−6 FV = \sqrt{\frac{2 \times 73 \, \mathrm{J}}{120 \times 10^{-6} \, \mathrm{F}}}
Key Concept
Energy stored in a capacitor
Explanation
The potential difference across a capacitor can be found by rearranging the formula for the energy stored in a capacitor and solving for VV.
91 Answer
The total potential at point P is −9.4×103 V-9.4 \times 10^{3} \, \mathrm{V}.
Solution
a
Use the formula for the potential due to a point charge at a distance rr: V=kQrV = \frac{kQ}{r}
b
Calculate the potential at point P due to each charge and add them up, considering their signs and distances from P
Key Concept
Superposition principle in electric potential
Explanation
The total electric potential at a point is the sum of the potentials due to each charge, taking into account their distances and signs.
92 Answer
The electric field intensity is 4.2×103 V/m4.2 \times 10^{3} \, \mathrm{V/m}, directed from A to B.
Solution
a
Use the formula for electric field due to a potential difference: E=ΔVdE = \frac{\Delta V}{d}, where EE is the electric field, ΔV\Delta V is the potential difference, and dd is the distance between the points
b
Substitute the given values: E=95 V−28 V0.016 mE = \frac{95 \, \mathrm{V} - 28 \, \mathrm{V}}{0.016 \, \mathrm{m}}
Key Concept
Electric field and potential difference
Explanation
The electric field intensity between two points is calculated by dividing the potential difference by the distance between the points. The direction is from higher to lower potential.
93 Answer
The potential difference VB−VAV_{B}-V_{A} is 88 V.
Solution
a
Use the formula for electric potential due to a point charge: V=kQrV = \frac{kQ}{r}
b
Calculate the potential at points A and B due to the charge and find the difference VB−VAV_{B}-V_{A}
Key Concept
Electric potential due to a point charge
Explanation
The difference in electric potential between two points in the field of a point charge is found by calculating the potential at each point and subtracting them.
94 Answer
The work done on the charge from A to B and from A to C is not provided in the question.
Solution
a
Use the work-energy theorem: W=qΔVW = q\Delta V, where WW is the work done, qq is the charge, and ΔV\Delta V is the potential difference
b
Calculate the work done for each path by finding the potential difference along the path and multiplying by the charge
Key Concept
Work done by electric force
Explanation
The work done on a charge by the electric force when moving between two points is the product of the charge and the potential difference between those points.
95 Answer
The dielectric constant εcircle\varepsilon_{\text{circle}} for the material between the circular plates is 3.82.
Solution
a
Use the formula for capacitance with dielectric: C=εε0AdC = \varepsilon \frac{\varepsilon_0 A}{d}, where CC is the capacitance, ε\varepsilon is the dielectric constant, ε0\varepsilon_0 is the vacuum permittivity, AA is the area of the plates, and dd is the distance between the plates
b
Since the capacitances are the same, set the capacitance formulas for the square and circular plates equal to each other and solve for εcircle\varepsilon_{\text{circle}}
Key Concept
Capacitance with dielectric
Explanation
The dielectric constant can be found by comparing the capacitance formulas for different geometries and solving for the unknown dielectric constant.
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