###### Advertisements

###### Advertisements

Two infinitely long straight parallel wires, '1' and '2', carrying steady currents I_{1} and I_{2} in the same direction are separated by a distance d. Obtain the expression for the magnetic field `vecB`due to the wire '1' acting on wire '2'. Hence find out, with the help of a suitable diagram, the magnitude and direction of this force per unit length on wire '2' due to wire '1'. How does the nature of this force changes if the currents are in opposite direction? Use this expression to define the S.I. unit of current.

###### Advertisements

#### Solution

Consider a straight conductor XY lying in the plane of paper. Consider a point P at a perpendicular distance *a* from straight conductor.

Magnetic field induction (*B*) at a point P due to current *I* passing through conductor XY is given by

`B=(μ0I)/(4πa)[sinϕ_1+sinϕ_2]`

where ϕ_{1} and ϕ_{2}_{ }are the angles made by point X and Y, respectively

At the centre of the infinite long wire, ϕ_{1}=ϕ_{2}=90°

`B=(μ_0I)/(4πa)[sin90^@+sin^@]`

`⇒B=μ_0/(4π) (2I)/a .....(1)`

Let 1 and 2 be two long infinite straight conductors. Let *I*_{1} and *I*_{2} be the current flowing through the conductor 1 and 2 and they are *d* distance apart from each other as shown in the figure.

The magnetic field induction (*B*) at a point P on conductor 2 due to current *I*_{1} passing through conductor 1 is given by

`B=(μ_02I_1)/(4πd) ` [From (1), where a=d]

According to right hand rule, the direction of this magnetic field is perpendicular to the plane of the paper inward.

Since the conductor 2 lies in this magnetic field of conductor 1, force experienced (*F*_{2}) by unit length of conductor 2 will be

F_{2}=B_{1}I_{2}×1=B_{1}I_{2}

`∴ F2=μ_0/(4π) (2I_1I_2)/d`

Conductor 1 also experiences the same amount of force, directed towards the conductor 2. Hence, conductor 1 and conductor 2 attract each other. Thus, two linear parallel conductors carrying currents in the same direction attract and repel each other, when the current flows in the opposite direction.

Let I_1=I_1=1A; r=1 m

Then,

`F_1=F_2=F=10^(−7) (2xx1xx1)/1`

`⇒F=2×10^(−7) N/m`

Thus, one ampere is that value of constant current which when flowing through each of the two parallel uniform long linear conductors placed in free space at a distance of 1 m from each other will attract or repel each other with a force of 2 × 10^{−7} N per metre of their length.

#### APPEARS IN

#### RELATED QUESTIONS

Derive the expression for force per unit length between two long straight parallel current carrying conductors. Hence define one ampere.

and ```vecE` and `vecB`denote electric and magnetic fields in a frame *S* and `vecE`→ and `vecB` in another frame *S*' moving with respect to *S* at a velocity `vecV` Two of the following equations are wrong. Identify them.

(a) `B_y^, = B_y + (vE_z)/c^2`

(b) `E_y^' = E_y - (vB_z)/(c^2)`

`(c) Ey = By + vE_z`

`(d) E_y = E_y + vB_z`

An electron is moving along the positive *x*-axis. You want to apply a magnetic field for a short time so that the electron may reverse its direction and move parallel to the negative *x*-axis. This can be done by applying the magnetic field along

(a) *y*-axis

(b) *z*-axis

(c) *y*-axis only

(d) *z*-axis only

Two parallel, long wires carry currents i_{1} and i_{2} with i_{1} > i_{2}. When the currents are in the same direction, the magnetic field at a point midway between the wires is 10 µT. If the direction of i_{2} is reversed, the field becomes 30 µT. The ratio i_{1}/i_{2} is

A current of 10 A is established in a long wire along the positive z-axis. Find the magnetic field \[\vec{B}\] at the point (1 m, 0, 0).

A long, straight wire of radius r carries a current i and is placed horizontally in a uniform magnetic field B pointing vertically upward. The current is uniformly distributed over its cross section. (a) At what points will the resultant magnetic field have maximum magnitude? What will be the maximum magnitude? (b) What will be the minimum magnitude of the resultant magnetic field?

A hypothetical magnetic field existing in a region is given by `vecB = B_0 vece` where `vece`_r denotes the unit vector along the radial direction. A circular loop of radius *a*, carrying a current *i*, is placed with its plane parallel to the *x*−*y* plane and the centre at (0, 0, *d*). Find the magnitude of the magnetic force acting on the loop.

A straight wire of length *l* can slide on two parallel plastic rails kept in a horizontal plane with a separation *d*. The coefficient of friction between the wire and the rails is µ. If the wire carries a current *i*, what minimum magnetic field should exist in the space in order to slide the wire on the rails?

The magnetic field existing in a region is given by `vecB = B_0(1 + x/1)veck` . A square loop of edge *l* and carrying a current *i*, is placed with its edges parallel to the *x*−*y* axes. Find the magnitude of the net magnetic force experienced by the loop.

A rectangular coil of 100 turns has length 5 cm and width 4 cm. It is placed with its plane parallel to a uniform magnetic field and a current of 2 A is sent through the coil. Find the magnitude of the magnetic field *B* if the torque acting on the coil is 0.2 N m^{−1}

Two parallel wires carry equal currents of 10 A along the same direction and are separated by a distance of 2.0 cm. Find the magnetic field at a point which is 2.0 cm away from each of these wires.

A long, straight wire carries a current i. Let B_{1} be the magnetic field at a point P at a distance d from the wire. Consider a section of length l of this wire such that the point P lies on a perpendicular bisector of the section B_{2} be the magnetic field at this point due to this second only. Find the value of d/l so that B_{2} differs from B_{1} by 1%.

Define Ampere in terms of force between two current carrying conductors.

**Answer the following question.**

Two infinitely long straight wire A_{1} and A_{2} carrying currents I and 2I flowing in the same direction are kept' distance apart. Where should a third straight wire A_{3} carrying current 1.5 I be placed between A_{1} and A_{2} so that it experiences no net force due to A_{1} and A_{2}? Does the net force acting on A_{3} depend on the current flowing through it?

If a current I is flowing in a straight wire parallel to x-axis and magnetic field is there in the y-axis then, ______.

According to Ampere's circuital law, ______.

Three infinitely long parallel straight current-carrying wires A, B and C are kept at equal distance from each other as shown in the figure. The wire C experiences net force F. The net force on wire C, when the current in wire A is reversed will be ______.

Equal currents are passing through two very long and straight parallel wires in the same direction. They will ______

Five long wires A, B, C, D and E, each carrying current I are arranged to form edges of a pentagonal prism as shown in figure. Each carries current out of the plane of paper.

- What will be magnetic induction at a point on the axis O? AxisE is at a distance R from each wire.
- What will be the field if current in one of the wires (say A) is switched off?
- What if current in one of the wire (say) A is reversed?

Two long parallel wires kept 2 m apart carry 3A current each, in the same direction. The force per unit length on one wire due to the other is ______.