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Magnetic Force on a Current-Carrying Conductor

Because charges ordinarily cannot escape a conductor, the magnetic force on charges moving in a conductor is transmitted to the conductor itself.

A diagram showing a circuit with current I running through it. One section of the wire passes between the north and south poles of a magnet with a diameter l. Magnetic field B is oriented toward the right, from the north to the south pole of the magnet, across the wire. The current runs out of the page. The force on the wire is directed up. An illustration of the right hand rule 1 shows the thumb pointing out of the page in the direction of the current, the fingers pointing right in the direction of B, and the F vector pointing up and away from the palm.

Figure 22.30 The magnetic field exerts a force on a current-carrying wire in a direction given by the right hand rule 1 (the same direction as that on the individual moving charges). This force can easily be large enough to move the wire, since typical currents consist of very large numbers of moving charges.

We can derive an expression for the magnetic force on a current by taking a sum of the magnetic forces on individual charges. (The forces add because they are in the same direction.) The force on an individual charge moving at the drift velocity $v_{d} v_{d}$ is given by $F = {qv}_{d} B sin θ F = {qv}_{d} B$ . Taking $B B size 12{B} {}$ to be uniform over a length of wire $l l$ and zero elsewhere, the total magnetic force on the wire is then $F = ({qv}_{d} B sin θ) (N) F = ({qv}_{d} B size 12{F= $ ital "qv" rSub { size 8{d} } B"sin"θ $ $ N $ } {}$ , where $N N size 12{N} {}$ is the number of charge carriers in the section of wire of length $l l size 12{l} {}$ . Now, $N = nV N = nV size 12{N= ital "nV"} {}$ , where $n n size 12{n} {}$ is the number of charge carriers per unit volume and $V V size 12{V} {}$ is the volume of wire in the field. Noting that $V = Al V = Al size 12{V= ital "Al"} {}$ , where $A A size 12{A} {}$ is the cross-sectional area of the wire, then the force on the wire is $F = ({qv}_{d} B sin θ) (nAl) F = ({qv}_{d} B$ . Gathering terms,

F = ({nqAv}_{d}) lB sin θ . F = ({nqAv}_{d}) lB . size 12{F= \(  ital "nqAv" rSub { size 8{d} }  \)  ital "lB""sin"θ} {}

22.15

Because ${nqAv}_{d} = I {nqAv}_{d} = I size 12{ ital "nqAv" rSub { size 8{d} } =I} {}$ (see Current),

F = IlB sin θ F = IlB sin size 12{F= ital "IlB""sin"θ} {}

22.16

is the equation for magnetic force on a length $l l$ of wire carrying a current $I I$ in a uniform magnetic field $B B$ , as shown in Figure 22.31. If we divide both sides of this expression by $l l$ , we find that the magnetic force per unit length of wire in a uniform field is $\frac{F}{l} = IB sin θ \frac{F}{l} = IB size 12{ { {F} over {l} } = ital "IB""sin"θ} {}$ . The direction of this force is given by RHR-1, with the thumb in the direction of the current $I I size 12{I} {}$ . Then, with the fingers in the direction of $B B size 12{B} {}$ , a perpendicular to the palm points in the direction of $F F size 12{F} {}$ , as in Figure 22.31.

Illustration of the right hand rule 1 showing the thumb pointing right in the direction of current I, the fingers pointing into the page with magnetic field B, and the force directed up, away from the palm.

Figure 22.31 The force on a current-carrying wire in a magnetic field is

F = IlB sin θ F = IlB sin size 12{F= ital "IlB""sin"θ} {}

. Its direction is given by RHR-1.

Example 22.4

Calculating Magnetic Force on a Current-Carrying Wire: A Strong Magnetic Field

Calculate the force on the wire shown in Figure 22.30, given $B = 1 . 50 T B = 1 . 50 T size 12{B=1 "." "50"" T"} {}$ , $l = 5 . 00 cm l = 5 . 00 cm size 12{l=5 "." "00"" cm"} {}$ , and $I = 20 . 0 A I = 20 . 0 size 12{I="20" "." 0 A} {}$ .

Strategy

The force can be found with the given information by using $F = IlB sin θ F = IlB size 12{F= ital "IlB""sin"θ} {}$ and noting that the angle $θ θ size 12{θ} {}$ between $I I size 12{I} {}$ and $B B size 12{B} {}$ is $90º 90º$ , so that $sin θ = 1 sin$ .

Solution

Entering the given values into $F = IlB sin θ F = IlB size 12{F= ital "IlB""sin"θ} {}$ yields

F = IlB sin θ = (20 .0 A) (0 . 0500 m) (1 . 50 T) (1) . F = IlB . size 12{F= ital "IlB""sin"θ= left ("20" "." 0" A" right ) left (0 "." "0500"" m" right ) left (1 "." "50"" T" right ) left (1 right )} {}

22.17

The units for tesla are $1 T = \frac{N}{A \cdot m} 1 T = \frac{N}{A \cdot m} size 12{"1 T"= { {N} over {A cdot m} } } {}$ ; thus,

F = 1 . 50 N. F = 1 . 50 N. size 12{F=1 "." "50"" N"} {}

22.18

Discussion

This large magnetic field creates a significant force on a small length of wire.

Magnetic force on current-carrying conductors is used to convert electric energy to work. (Motors are a prime example—they employ loops of wire and are considered in the next section.) Magnetohydrodynamics (MHD) is the technical name given to a clever application where magnetic force pumps fluids without moving mechanical parts. (See Figure 22.32.)

Diagram showing a cylinder of fluid of diameter l placed between the north and south poles of a magnet. The north pole is to the left. The south pole is to the right. The cylinder is oriented out of the page. The magnetic field is oriented toward the right, from the north to the south pole, and across the cylinder of fluid. A current-carrying wire runs through the fluid cylinder with current I oriented downward, perpendicular to the cylinder. Negative charges within the fluid have a velocity vector pointing up. Positive charges within the fluid have a velocity vector pointing downward. The force on the fluid is out of the page. An illustration of the right hand rule 1 shows the thumb pointing downward with the current, the fingers pointing to the right with B, and force F oriented out of the page, away from the palm.

Figure 22.32 Magnetohydrodynamics. The magnetic force on the current passed through this fluid can be used as a nonmechanical pump.

A strong magnetic field is applied across a tube and a current is passed through the fluid at right angles to the field, resulting in a force on the fluid parallel to the tube axis as shown. The absence of moving parts makes this attractive for moving a hot, chemically active substance, such as the liquid sodium employed in some nuclear reactors. Experimental artificial hearts are testing with this technique for pumping blood, perhaps circumventing the adverse effects of mechanical pumps. (Cell membranes, however, are affected by the large fields needed in MHD, delaying its practical application in humans.) MHD propulsion for nuclear submarines has been proposed, because it could be considerably quieter than conventional propeller drives. The deterrent value of nuclear submarines is based on their ability to hide and survive a first or second nuclear strike. As we slowly disassemble our nuclear weapons arsenals, the submarine branch will be the last to be decommissioned because of this ability (See Figure 22.33.) Existing MHD drives are heavy and inefficient—much development work is needed.

Diagram showing a zoom in to a magnetohydrodynamic propulsion system on a nuclear submarine. Liquid moves through the thruster duct, which is oriented out of the page. Magnetic fields emanate from the coils and pass through a duct. The magnetic flux is oriented up, perpendicular to the duct. Each duct is wrapped in saddle-shaped superconducting coils. An electric current runs to the right, through the liquid and perpendicular to the velocity of the liquid. The electric current flows between a pair of electrodes inside each thruster duct. A repulsive interaction between the magnetic field and electric current drives water through the duct. An illustration of the right hand rule shows the thumb pointing to the right with the electric current. The fingers point up with the magnetic field. The force on the liquid is oriented out of the page, away from the palm.

Figure 22.33 An MHD propulsion system in a nuclear submarine could produce significantly less turbulence than propellers and allow it to run more silently. The development of a silent drive submarine was dramatized in the book and the film The Hunt for Red October.