At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are 4.984.984.98 V and 16.216.216.2 V/m, respectively. (Take V=0V = 0V=0 at infinity.) What is the distance to the point charge?

Start by recalling the relationship between electric potential (V) and electric field (E) for a point charge. The electric potential at a distance r from a point charge Q is given by the formula: V=kQr, where k is Coulomb's constant (approximately $$8.99 × 10^9 N m$$²/C²). The electric field magnitude at the same distance r from the point charge is given by: E=kQr2. You have two equations: V=kQr and E=kQr2. You can solve these simultaneously to find the distance r. First, express Q in terms of V and r from the first equation: Q=Vrk. Substitute the expression for Q into the equation for E: E=kVrkr2. Simplify this to get: E=Vr. Rearrange the equation E=Vr to solve for r: r=VE. Substitute the given values for V and E to find the distance r.

Ch 23: Electric Potential

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 23: Electric Potential

Problema 22a

Capitolo 23, Problema 22a

At a certain distance from a point charge, the potential and electric-field magnitude due to that charge are $4.98$ V and $16.2$ V/m, respectively. (Take $V = 0$ at infinity.) What is the distance to the point charge?

Guida verificata passo dopo passo

Start by recalling the relationship between electric potential (V) and electric field (E) for a point charge. The electric potential at a distance r from a point charge Q is given by the formula:

V = \frac{k Q}{r}

, where k is Coulomb's constant (approximately 8.99 × 10^9 N m²/C²).

The electric field magnitude at the same distance r from the point charge is given by:

E = \frac{k Q}{r^{2}}

You have two equations:

V = \frac{k Q}{r}

and

E = \frac{k Q}{r^{2}}

. You can solve these simultaneously to find the distance r. First, express Q in terms of V and r from the first equation:

Q = \frac{V r}{k}

Substitute the expression for Q into the equation for E:

E = \frac{k \frac{V r}{k}}{r^{2}}

. Simplify this to get:

E = \frac{V}{r}

Rearrange the equation

E = \frac{V}{r}

to solve for r:

r = \frac{V}{E}

. Substitute the given values for V and E to find the distance r.

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Electric Potential

Electric potential (V) at a point in space is the work done per unit charge in bringing a positive test charge from infinity to that point. It is a scalar quantity and is measured in volts (V). For a point charge, the potential is given by V = kQ/r, where k is Coulomb's constant, Q is the charge, and r is the distance from the charge.

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Electric Potential

Electric Field

The electric field (E) is a vector field around a charged object where a force would be exerted on other charges. It is defined as the force per unit charge and is measured in volts per meter (V/m). For a point charge, the electric field is given by E = kQ/r², indicating how the field strength decreases with the square of the distance from the charge.

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Intro to Electric Fields

Relationship Between Electric Field and Potential

The electric field is related to the electric potential by the gradient, where E = -dV/dr for a radial field. This relationship shows that the electric field is the rate of change of the potential with respect to distance. For a point charge, this can be used to derive the distance from the charge when both the potential and field magnitude are known.

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Percorso guidato

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Relationships Between Force, Field, Energy, Potential

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