What is the current in a wire of radius R = 2 . 67 mm if the magnitude of the current density is given by (a) J a = J 0 r R A m - 2 and (b) J b = J 0 ( 1 - r R ) A m - 2 , in which r is the radial distance in m and J 0 = 5 . 50 × 10 4 A m - 2 ? (c) Which function maximises the current density near the wire's surface?

In this question we have been given respective values of current densities for two different values of current densities. So, for the first case in which the value of current density is J a = J 0 r R . We know that the relation of current density in terms of current density as, I = ∫ J d A where, J is the current density and A = π r 2 ⇒ d A = 2 π r d r (a) I a = ∫ 0 R J a 2 π r d r ∴ I a = ∫ 0 R J 0 r R 2 π r d r = 2 π J 0 R ∫ 0 R r 2 d r ∴ I a = 2 π J 0 R r 3 3 o R = 2 π J 0 R R 3 3 ∴ I a = 2 π J 0 R 2 3 Putting the values, J 0 = 5 . 50 × 10 4 A m - 2 and R = 2 . 67 mm = 2 . 67 × 10 - 3 m ,we get, I a = 2 × 3 . 14 × 5 . 50 × 10 4 × 2 . 67 × 10 - 3 2 3 ∴ I a = 1 . 332 A Now for the second case, I b = ∫ 0 R J b 2 π r d r ∵ A = π r 2 ⇒ d A = 2 π r d r Here, J b = J 0 ( 1 - r R ) , I b = ∫ 0 R J 0 1 - r / R 2 πrdr = 2 π J 0 ∫ 0 R r - r 2 R d r ⇒ I b = 2 π J 0 r 2 2 - r 3 3 R 0 R = 2 π J 0 R 2 2 - R 2 3 ∴ I b = π J 0 R 2 3 Putting the values J 0 = 5 . 50 × 10 4 A m - 2 and R = 2 . 67 mm = 2 . 67 × 10 - 3 m we get, I b = 0 . 666 A (c) The result is not same as that in part (a) because J b is higher near the centre of the cylinder (where the area is smaller for the same radial interval) and lower outward, which results in a lower average current density over the cross section and consequently a lower current than that in part (a). So, J a has its maximum value near the surface of the wire.

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The (United States) National Electric Code, which sets maximum safe currents for insulated copper wires of various diameters, is given (in part) in the table. (a) Plot the safe current density as a function of diameter. Which wire gauge has the maximum safe current density? (Gauge is a way of identifying wire diameters, and $1 mil = 10^{- 3}$ $in$ .) (b) What is the current density (assumed to be uniform in $8$ -gauge wire for a current of $35 A ?$

$\begin{array}{lrrrrrrrr} Gauge & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 \\ Diameter, mils & 204 & 162 & 129 & 102 & 81 & 64 & 51 & 40 \\ Safe current, A & 70 & 50 & 35 & 25 & 20 & 15 & 6 & 3 \end{array}$

Important Questions on Current and Resistance

HARD

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 26 - Current and Resistance>Problems>Q 45

What is the current in a wire of radius

R = 2.67 mm

if the magnitude of the current density is given by (a)

J_{a} = \frac{J_{0} r}{R} A m^{- 2}

and (b)

J_{b} = J_{0} (1 - \frac{r}{R}) A m^{- 2},

in which

r

is the radial distance in

m

and

J_{0} = 5.50 \times 10^{4} A m^{- 2} ?

HARD

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 26 - Current and Resistance>Problems>Q 46

A certain cylindrical wire carries current. We draw a circle of radius $r$ around its central axis in figure (a) to determine the current $i$ within the circle. Figure (b) shows current $i$ as a function of $r^{2}$ . The vertical scale is set by $i_{s} = 4.0 mA,$ and the horizontal scale is set by $r_{s}^{2} = 8.0 {mm}^{2} .$ (a) Is the current density uniform? (b) If so, what is its magnitude? (c) What is the current between $r = 0$ and $r = 2.0 mm ?$

$(a)$ Question Image

$(b)$ Question Image

EASY

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IMPORTANT

Principles Of Physics International Student Version>Chapter 26 - Current and Resistance>Problems>Q 47

What is the resistance of a wire with radius

0.500 mm

and length

4.3 m,

made from a material with resistivity

2.0 \times 10^{- 8} Ω m ?

MEDIUM

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 26 - Current and Resistance>Problems>Q 48

Figure (a) gives the magnitude $E (x)$ of the electric fields that have been set up by a battery along a resistive rod of length $9.00 mm$ (Fig.b). The vertical scale is set by $E_{s} = 8.00 \times 10^{3} V / m$ The rod consists of three sections of the same material but with different radii. (The schematic diagram of Fig.(b) does not indicate the different radii.) The radius of section $3$ is $1.70 mm$ . What is the radius of (a) section $1$ and (b) section $2 ?$

$(a)$ Question Image

$(b)$ Question Image

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JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 26 - Current and Resistance>Problems>Q 49

Two conductors are made of the same material and have the same length. Conductor

A

is a solid wire of radius

1.0 mm

. Conductor

B

is a hollow tube of outside radius

2.2 mm

and inside radius

1.0 mm .

What is the resistance ratio

\frac{R_{A}}{R_{B}},

measured between their ends?

MEDIUM

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 26 - Current and Resistance>Problems>Q 50

Near the Earth, the density of protons in the solar wind (a stream of particles from the Sun) can be

4.63 {cm}^{- 3},

and their speed can be

391 km s^{- 1}

. (a) Find the current density of these protons. (b) If Earth's magnetic field did not deflect the protons, what total current would Earth receive?

MEDIUM

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 26 - Current and Resistance>Problems>Q 51

A common flashlight bulb is rated at $0.20 A$ and $3.2 V$ (the values of the current and voltage under operating conditions). If the resistance of the tungsten bulb filament at room temperature $(20 ° C)$ is $1.1 Ω,$ what is the temperature of the filament when the bulb is on? Temperature coefficient of resistance of filament is $α = 4.5 \times 10^{- 3} K^{- 1}$ .

MEDIUM

JEE Main

IMPORTANT

Principles Of Physics International Student Version>Chapter 26 - Current and Resistance>Problems>Q 52

An isolated conducting sphere has a

20 cm

radius. One wire carries a current of

1.0000020 A

into it. Another wire carries a current of

1.0000000 A

out of it. How long would it take for the sphere to increase in potential by

1000 V ?

Important Questions on Current and Resistance

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