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Chapter 22: Problem 18

Show that \(1 \mathrm{V} / \mathrm{m}\) is the same as \(1 \mathrm{N} / \mathrm{C}\)

Short Answer

Expert verified
Yes, 1V/m is indeed equivalent to 1N/C. The electric field can be represented using either of these units, as shown by the formulas for electric field and electric potential difference.

Step by step solution

01

Understanding Electric Field and Electric Potential Difference

The electric field (E) is a vector quantity that represents the force (F) per unit charge (Q) that would be experienced by a stationary test charge placed in it. It's expressed in N/C. This is represented by the formula: \(E = \frac{F}{Q}\). The electric potential difference (V), also known as voltage, is the work done (W) per unit charge (Q). This is usually expressed in volts (V). The formula for this is: \(V = \frac{W}{Q}\). In the context of electric fields, the work done is equal to the force times the distance (d), or \(W = F \cdot d \). Therefore, the formula for electric potential difference in terms of force and distance is \(V = \frac{F \cdot d}{Q}\)
02

Relating Electric Field and Electric Potential Difference

If we rearrange the formula for the electric potential difference in terms of force and distance (\(V = \frac{F \cdot d}{Q}\)), we see that it can be expressed as the product of force per unit charge (the electric field, E) and distance: \(V = E \cdot d\)Now, if we solve this equation for the electric field (E), we get \(E = \frac{V}{d}\). This shows that the electric field in terms of voltage and distance is expressed in volts per meter (V/m).
03

Proving the Equivalence of the Units

We have shown that the unit for electric field can be expressed as either N/C (from \(E = \frac{F}{Q}\)) or V/m (from \(E = \frac{V}{d}\)). Therefore, we have effectively proven that 1 N/C is equivalent to 1 V/m.

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Most popular questions from this chapter

A conducting sphere \(5.0 \mathrm{cm}\) in radius carries \(60 \mathrm{nC} .\) It's surrounded by a concentric spherical conducting shell of radius \(15 \mathrm{cm}\) carrying -60 nC. (a) Find the potential at the sphere's surface, taking \(V=0\) at infinity. (b) Repeat for the case when the shell carries \(+60 \mathrm{nC}\)

Proton-beam therapy can be preferable to X rays for cancer treatment (although much more expensive) because protons deliver most of their energy to the tumor, with less damage to healthy tissue. A cyclotron used to accelerate protons for cancer treatment repeatedly passes the protons through a 15 -kV potential difference. (a) How many passes are needed to bring the protons' kinetic energy to \(1.2 \times 10^{-11} \mathrm{J} ?\) (b) What's that energy in eV?

Why can a bird perch on a high-voltage power line without getting electrocuted?

Two equal but opposite charges form a dipole. Describe the equipotential surface on which \(V=0\)

A sphere of radius \(R\) carries negative charge of magnitude \(Q,\) distributed in a spherically symmetric way. Find an expression for the escape speed for a proton at the sphere's surface-that is, the speed that would enable the proton to escape to arbitrarily large distances starting at the sphere's surface.
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