Examville.com - Magnetic Field Due To Elec Current

Magnetic Field Due To Electric Current 1. The area around a current carrying conductor or a magnet, where the magnetic effects can be experienced is called a magnetic field. 2. The direction of magnetic field at any given point is taken to be the direction in which a north pole would move if placed in that position. The path which such a pole would follow is called a magnetic field line (line of force). 3. The magnetic lines of force emerge out from North Pole, pass through the surrounding medium, and then again enter the South Pole. We can say that, magnetic lines of force form closed loops. 4. Irrespective of the shape of the current carrying conductor, magnetic lines of force always form closed loops. 5. The magnetic flux density is a measure of magnetic field concentration, or number of magnetic lines of force in each sq. meter of the field. 6. The magnitude and direction of magnetic field represented by magnetic flux density (B), it is also called magnetic induction. The direction of B at a point is that of the tangent to the magnetic line of force at that point. 7. The S.I unit of magnetic flux density is tesla (given by T), and its c.g.s unit is gauss (given by G). 1 Tesla = 1 Wb / m2 = 104G

8. The magnetic flux (ΦB) through a region is the number of magnetic lines of force passing normally through the region. Magnetic flux, ΦB = Area (A) x component of B along normal to the area = AB cosθ = A.B ΦB= A.B So, magnetic flux through an area is the dot or scalar product of A and B. So, ΦB is a scalar quantity. 9. It has been found out by experiments that magnetic force on a charge q moving with velocity v in a uniform external magnetic field B is given by Fm = q x v x Bsinθ, where θ is the angle between v and B Fm = q (v x B) As, Fm is perpendicular to the plane containing v and B. (a). Fm = 0, when θ = 00 or 1800. Fm is also zero when v is zero. When θ= 900, Fm = qvB (maximum value) (b). As Fm is always perpendicular to v, a uniform magnetic field can do no work on a moving charge. (c). A uniform magnetic field will cause a moving charged particle of mass m to move in a circle of radius r given by (mv2) / r = qvB Or,

r = (mv) / qB (as θ = 900)

10. We can find the direction of magnetic force (Fm) on a moving charge in a magnetic field by right hand rule for cross product, Put your right hand in a direction, so that your outstretched fingers point along the direction of motion of positively charged particle. The direction of your hand should be such that when you bend your fingers, they must point along the direction of magnetic field (B), then your extended thumb will point in a direction of force on the charged particle. 11. According to Biot Savart Law, The magnitude of magnetic flux density dB at a point P which is at a distance r from a very small length dl of a conductor carrying current I is given by dB α (I dl sinθ) / r2 Or,

dB = K (I dl sinθ)/ r2

Where θ is the angle between the small length dl and the line joining it to point P, K is the constant of proportionality. It depends on the medium around the conductor. In a vacuum or air K = μ0 / 4π. dB = (μ0 / 4π)( [ I dl sinθ] / [r2] ) In the vector form dB = (μ0 / 4π)( I [dl x r] / [r3] ) (a). Biot savart law holds strictly for steady currents. (b). Idl is called current element. (c). The direction of dB is perpendicular to the plane containing dl and r. It can be found by right hand rule for cross product. (d). Biot savart law can not be tested directly as it is not possible to have a current carrying conductor of length dl. But it can be used to derive expressions for flux densities of real conductors

and these give values which are in agreement with those determined by experiments. 12. A current carrying loop behaves as a magnetic dipole. The magnitude of magnetic dipole moment (M) of current carrying loop is given by m=nIA Where n = number of turns of loop, I = current, A = area of loop. 13. The magnitude of magnetic field at the centre of a current carrying circular coil of radius r is given by B = (μ0I) / 2r, where I is the current in the coil. If the coil has n number of turns, each carrying current in the same direction, then B = (μ0nI) / 2r 14. The magnitude of magnetic field at a point P located at a perpendicular distance a from a finite straight conductor carrying current I is B = (μ0I / 4πa) [sinΦ2 + sinΦ1] If the conductor is infinitely long, then Φ1 = Φ2 = π / 2 So, B = (μ0 / 4π) (I / a) (sinπ/2 + sinπ/2) = (μ0 / 4π) (2I / a)

15. Let us consider a circular coil of radius r, centre O and current I, Let us assume that plane of the coil be perpendicular to plane of the paper, then the magnitude of the field at P on the axis of the coil (if OP = r) is B = (μ0 n I a2) / 2(a2 + r2)3/2 along PX Where, n = number of turns of the coil.

(a). If p is at the centre of coil, r = 0, then B becomes B = (μ0 n I) / 2a, at the centre of the coil. b). If the point P is far away from the centre of the coil, r >a, then B = (μ0 n I a2) / (2r3) 16. According to Ampere’s circuital law, the line integral of magnetic field B around any closed path in vacuum/air is equal to μ0 times the total current enclosed by that path. (a). Ampere’s law is true only for steady currents. (b). This law is an alternative to Biot- stavart law, but it can be used only in those situations where it is impossible to find the line integral of chosen path.

17. The magnitude of magnetic field on the axis of a long air cored solenoid can be given by B = μ0 n I Where n = number of turns per meter. (a). B is directed along the axis of the solenoid. (b). At the either of the ends of the solenoid, B along the axis is (μ0 n I) / 2 (c). The magnetic field outside a solenoid is zero. 18. The magnitude of magnetic field due to air cored toroid is B = μ0 n I. (a). the expression is same for solenoid because a toroid is a solenoid in the form of a ring. (b). Magnetic field is only confined to inside the toroid. (c). The magnetic field outside toroid is zero. (d).The magnetic field inside the toroid is constant and is always tangent to the field lines. 19. The magnetic field produced by current carrying conductor arrangement is always perpendicular to the plane of the conductor arrangement. Let us assume that a current carrying coil is in the vertical plane, then the magnetic field produced by it will be in the horizontal plane and vice – versa.

20. We can find the direction of magnetic field due to various current carrying conductor arrangements by using the following rules, (a). Right hand palm rule: We can find the direction of magnetic field at the centre of a current carrying circular coil by right hand palm rule, which states as follows, Orient the thumb of your right hand perpendicular to the grip of the fingers such that curvature of the fingers point in the direction of current in the circular coil. Then thumb will point in the direction of magnetic field lines near the centre of the circular coil. (b). Right hand grip rule: If we have a straight current carrying wire, the magnetic field lines can be found by using right hand grip rule, which states as followsGrip the wire with your right hand with thumb pointing in the direction of the conventional current. Then curled fingers point in the direction of the magnetic field lines. (c). Right hand rule for solenoid: We can find the direction of magnetic field due to a current carrying solenoid by using right rule for a coil of several turns which states as followsGraph the solenoid with right hand so that fingers are curled in the direction of current, Then thumb stretched parallel to the axis of the solenoid will point towards the N-pole end of the solenoid. (d). Right hand fist rule: The magnetic field at the centre of a circular coil is along the axis of the coil. We can find the direction of magnetic field by right hand fist rule which states as followsHold the axis of the coil in the right hand fist in such a way that fingers point in the direction of current in the coil. Then outstretched thumb gives the direction of the magnetic field.

We can apply all the above rules in reverse. If we know the direction of magnetic field due to current carrying conductor arrangement, we can determine the direction of current in the coil / conductor. 21. The magnetic field produced by current carrying conductor arrangement is always perpendicular to the plane of the conductor arrangement. Let us assume that a current carrying coil is in the vertical plane, then the magnetic field produced by it will be in the horizontal plane and vice – versa. 22. Fm, the force on a moving charge in a magnetic field is perpendicular to the plane containing v and B. Right hand rule for cross product can be used to determine the direction of Fm. Right hand rule for cross product states that, Orient your hand so that outstretched fingers point along the motion of the positively charged particle, the orientation should be such that when you bend your fingers, they must point along the direction of magnetic field B. Then your extended thumb will point in the direction of force on the charged particle. We can apply right hand rule for cross product on an electron entering (negatively charged particle) the magnetic field. The direction of positive charge will be exactly opposite. When we apply right hand rule for cross product, we find that direction of force on the electron will be vertically upward. For a positively charged particle, it will be vertically downward. We note that the electric field always does not exert force on an electric charge which points in the direction parallel to the field, Magnetic force point in the direction parallel to magnetic field nor does it point in the direction of charge’s velocity. The magnetic force is perpendicular to both the magnetic field and the charge’s velocity.

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