Practice to calculate the distance of point of zero magnetic field between and outside the two parallel wires.

**Case 1**

- Consider two parallel current carrying conductors, as shown in figure.
- The direction of current in both the conductors is same and both conductors are of length \(\ell\), where \(\ell\) is very large.

- Force acting on wire 1 due to magnetic field of wire 2 is shown in figure.
- The magnetic force on wire 1 due to magnetic field of wire 2

\(\vec{F_{1/2}} = I_1\;\vec{l} × \vec {B_2}\)

where \(I_1\) is current in wire 1

\(B_2\) is magnetic field due to current in 2

Note:- \(\vec{l}\) is always taken in the direction of current flow.

- The magnetic force on wire 2 due to magnetic field of wire 1

\(\vec{F_{2/1}} = I_2\;\vec{l} × \vec {B_1}\)

**Case 2**

- Consider two conductors carrying current in opposite direction, as shown in figure.

- Force acting on wire 1 due to magnetic field of wire 2

\(\vec{F_{1/2}} = I_1\;\vec{l} × \vec {B_2}\)

- Force acting on wire 2 due to magnetic field of wire 1

\(\vec{F_{2/1}} = I_2\;\vec{l} × \vec {B_1}\)

**Conclusion: **Parallel conductors carrying current in the same direction attract each other and parallel conductors carrying current in opposite direction repel each other.

- Consider two straight infinite conductors, 1 and 2, carrying current \(I_1\) and \(I_2\) respectively in the same direction as shown in figure.
- The magnetic force on length ( \(\ell\)) of wire 1 due to wire 2

\(\vec{F_{1/2}} = I_1\vec{\ell} × \vec{B_2}\)

where

\(\vec{\ell}\) is vector length of wire 1

\(\vec{B_2}\) is magnetic field vector of wire 1 due to wire 2

So, \(F_1 = I_1 {\ell}B_2\)

\(\Rightarrow F_1 = I_1 {\ell}(\dfrac{\mu_0I_2}{2 \pi a}) \;\;\;\;\;[\therefore \;B_2 = \dfrac{\mu_0I_2}{2 \pi a}\,\, for\,\,long\,\,wire]\)

\(\Rightarrow F_1 = \dfrac{\mu_0 I_ 1 I_2}{2 \pi a}{\ell}\)

- Force per unit length of wire 1 due to wire 2

\(\dfrac{F_1}{\ell} = \dfrac{\mu_0I_1I_2}{2\pi a}\)

- The point of zero magnetic field between two parallel current carrying wires, is possible only when current in both the wires is in same direction.
- If current in both the wires is in opposite direction, then the magnetic field due to both wires will be in same direction. So, point of zero magnetic field between both the wires is not possible.
**Case 1 :**In the shown figure, the direction of current \(I_1\) and \(I_2\) are in opposite direction, so, the magnetic field due to both wires is inside the page using right hand thumb rule.

**Case 2 :**In the shown figure, the direction of current \(I_1\) and \(I_2\) are in opposite direction, so, the magnetic field due to both wires is outside the page using right hand thumb rule.

- To calculate point of zero magnetic field, direction of current in both the wires should be same.
- Consider two infinitely long wires, placed at a distance 'd' apart, carrying current of \(I_1\) and \(I_2\) in the same direction, as shown in figure.
- Consider point P as the point of zero magnetic field at a perpendicular distance x from wire 1.
- Then, the magnetic field at P will be of same magnitude.

\(|\vec{B_{P_1}}|\) = \(|\vec{B_{P_2}}|\)

\(\Rightarrow \dfrac{\mu _0I_1}{2 \pi x}\;\;=\;\;\dfrac{\mu_0I_2}{2 \pi (d-x)}\)

\(\Rightarrow I_1(d-x) = I_2 x\)

\(\Rightarrow I_1d = (I_1+I_2)x\)

\(\Rightarrow x = \dfrac{I_1d}{(I_1+I_2)}\)

- The point of zero magnetic field outside the two infinite long parallel wires can be calculated if current in both the wires is in opposite direction.
- When the current through both wires is in same direction, point of zero magnetic field is not possible outside the wires as direction of magnetic field due to both the wires is same.
**Case 1:**In the shown figure, the direction of current in both wires \(I_1\) and \(I_2\) are in same direction. The direction of magnetic field to the left of wire 1 is outside the page due to both wires. Also, the direction of magnetic field at the right of wire 2 is inside the page due to both wires. Hence, point of zero magnetic field is not possible outside the wires.

**Case 2:**In the shown figure, the direction of current in both wires \(I_1\) and \(I_2\) are in same direction. The direction of magnetic field to the left of wire 1 is inside the page due to both wires. Also, the direction of magnetic field at the right of wire 2 is outside the page due to both wires. Hence, point of zero magnetic field is not possible outside the wires.

- To calculate point of zero magnetic field outside the wire, consider two infinitely long wires carrying current \(I_1\) and \(I_2\) in the opposite direction, placed at a distance \(d\) apart, as shown in figure.

- Consider a point P as point of zero magnetic field beyond the wire 1 at a perpendicular distance \(x\) as shown in figure.

\(|\vec{B_{p_1}}|\;=\;\ |\vec{B_{p_2}}|\)

\(\Rightarrow \dfrac{\mu_0I_1}{2 \pi x}\;\;=\;\;\dfrac{\mu_0I_2}{2\pi(d+x)}\)

\(\Rightarrow I_1(d+x) = I_2 x\)

\(x = (\dfrac{I_1}{I_2 - I_1}) \;d\)

- \((I_2 - I_1) \) should be greater than zero as \(x\) is a distance.

or, \(I_2 - I_1 >0\)

\(\Rightarrow I_2 >I_1\)

- It means point of zero magnetic field is possible beyond the wire of less current.
- If current in both the wires, \(I_1 \) and \(I_2\) is same, then
- Point of zero magnetic field

\(x = (\dfrac{I_1}{I_2 - I_1}) \;d\)

or, \(x = \dfrac {Id}{I-I} = \dfrac{Id}{0} = \infty\)

Thus, point of zero magnetic field is not possible.

A 2 cm at right of wire 2

B 10 cm at right of wire 2

C 6 cm at left of wire 2

D Can't be determined

A 3 cm, 8 A

B 5 cm, 3 A

C 2 cm, 0.67 A

D 2 cm, 3 A

A \(\dfrac{mg\pi\ell_1}{\mu_0\pi\ell}\)

B \(\dfrac{mgI_1}{I\ell}\)

C \(\dfrac{mg \;2\pi\;\ell_1}{\sqrt3 \;\mu_0\; I \;\ell}\)

D \(\dfrac {mg\;I\;\ell_1}{I_1 \ell}\)