EXHIBIT B

Exhibit 10.2

EXHIBIT B

WARRANT

to Purchase up to 1,881,809 Shares of the

Common Stock, $.01 Par Value Per Share,

of

COLUMBIA BANCORP

This is to certify that, for value received, Fulton Financial Corporation (“Fulton”) or any permitted transferee (Fulton or such transferee being hereinafter called the “Holder”) is entitled to purchase, subject to the provisions of this Warrant, from Columbia Bancorp, a Maryland corporation (“Columbia”), at any time on or after the date hereof, an aggregate of up to 1,881,809 fully paid and non-assessable shares of common stock, $.01 par value (the “Common Stock”), of Columbia at a price per share equal to $37.26, subject to adjustment as herein provided (the “Exercise Price”).

1. Exercise of Warrant. Subject to the provisions hereof and the limitations set forth in Paragraphs 2 and 9 of a Warrant Agreement of even date herewith by and between Fulton and Columbia (the “Warrant Agreement”), which Warrant Agreement was entered pursuant to an Agreement and Plan of Merger dated July 26, 2005, between Fulton and Columbia (the “Merger Agreement”), this Warrant may be exercised in whole or in part or sold, assigned or transferred at any time or from time to time on or after the date hereof. This Warrant shall be exercised by presentation and surrender hereof to Columbia at the principal office of Columbia, accompanied by (i) a written notice of exercise, (ii) payment to Columbia, for the account of Columbia, of the Exercise Price for the number of shares of Common Stock specified in such notice, and (iii) a certificate of the Holder specifying the event or events which have occurred and entitle the Holder to exercise this Warrant. The Exercise Price for the number of shares of Common Stock specified in the notice shall be payable in immediately available funds.

Upon such presentation and surrender, Columbia shall issue promptly (and within one business day if reasonably requested by the Holder) to the Holder or its assignee, transferee or designee the number of shares of Common Stock to which the Holder is entitled hereunder. Columbia covenants and warrants that such shares of Common Stock, when so issued, will be duly authorized, validly issued, fully paid and non-assessable, and free and clear of all liens and encumbrances.

If this Warrant is exercised in part only, Columbia shall, upon surrender of this Warrant for cancellation, execute and deliver a new Warrant evidencing the rights of the Holder thereof to purchase the balance of the shares of Common Stock issuable hereunder. Upon receipt by Columbia of this Warrant, in proper form for exercise, and subject to the limitations set forth in Paragraphs 2 and 9 of the Warrant Agreement, the Holder shall be deemed to be the holder of record of the shares of Common Stock issuable upon such exercise, notwithstanding that the

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stock transfer books of Columbia may then be closed or that certificates representing such shares of Common Stock shall not then be actually delivered to the Holder. Columbia shall pay all expenses, and any and all United States federal, state and local taxes and other charges, that may be payable in connection with the preparation, issuance and delivery of stock certificates pursuant to this Paragraph 1 in the name of the Holder or its assignee, transferee or designee.

2. Reservation of Shares; Preservation of Rights of Holder.

Columbia shall at all times while this Warrant is outstanding and unexercised, maintain and reserve, free from preemptive rights, such number of authorized but unissued shares of Common Stock as may be necessary so that this Warrant may be exercised without any additional authorization of Common Stock after giving effect to all other options, warrants, convertible securities and other rights to acquire shares of Common Stock at the time outstanding. Columbia further agrees that (i) it will not, by charter amendment or through reorganization, consolidation, merger, dissolution or sale of assets, or by any other voluntary act or omission, avoid or seek to avoid the observance or performance of any of the covenants, stipulations or conditions to be observed or performed hereunder or under the Warrant Agreement by Columbia, (ii) it will promptly take all action (including (A) complying with all pre-merger notification, reporting and waiting period requirements specified in 15 U.S.C. §18a and the regulations promulgated thereunder and (B) in the event that, under Section 3 of the Bank Holding Company Act of 1956, as amended (12 U.S.C. §1842(a)(3)), or the Change in Bank Control Act of 1978, as amended (12 U.S.C. §1817(j)), prior approval of the Board of Governors of the Federal Reserve System (the “Board”) is necessary before this Warrant may be exercised, cooperating fully with the Holder in preparing any and all such applications and providing such information to the Board as the Board may require) in order to permit the Holder to exercise this Warrant and Columbia duly and effectively to issue shares of its Common Stock hereunder, and (iii) it will promptly take all action reasonably necessary to protect the rights of the Holder against dilution as provided herein.

3. Fractional Shares. Columbia shall not be required to issue any fractional shares of Common Stock upon exercise of this Warrant. In lieu of any fractional shares, the Holder shall be entitled to receive an amount in cash equal to the amount of such fraction multiplied by the Exercise Price.

4. Exchange or Loss of Warrant. This Warrant is exchangeable, without expense, at the option of the Holder, upon presentation and surrender hereof at the principal office of Columbia for other warrants of different denominations entitling the Holder to purchase in the aggregate the same number of shares of Common Stock issuable hereunder. The term “Warrant” as used herein includes any warrants for which this Warrant may be exchanged. Upon receipt by Columbia of evidence reasonably satisfactory to it of the loss, theft, destruction or mutilation of this Warrant, and (in the case of loss, theft or destruction) of reasonably satisfactory indemnification, and upon surrender and cancellation of this Warrant, if mutilated, Columbia will execute and deliver a new Warrant of like tenor and date.

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5. Repurchase. The Holder shall have the right to require Columbia to repurchase all or any portion of this Warrant under the terms and subject to the conditions of Paragraph 7 of the Warrant Agreement.

6. Adjustment. The number of shares of Common Stock issuable upon the exercise of this Warrant and the Exercise Price shall be subject to adjustment from time to time as provided in this Paragraph 6.

(A) Stock Dividends, etc.

(1) Stock Dividends. In case Columbia shall pay or make a dividend or other distribution on any class of capital stock of Columbia payable in Common Stock, the number of shares of Common Stock issuable upon exercise of this Warrant shall be increased by multiplying such number of shares by a fraction of which the denominator shall be the number of shares of Common Stock outstanding at the close of business on the day immediately preceding the date of such distribution and the numerator shall be the sum of such number of shares and the total number of shares of Common Stock constituting such dividend or other distribution, such increase to become effective immediately after the opening of business on the day following such distribution.

(2) Subdivisions. In case outstanding shares of Common Stock shall be subdivided into a greater number of shares of Common Stock, the number of shares of Common Stock issuable upon exercise of this Warrant at the opening of business on the day following the day upon which such subdivision becomes effective shall be proportionately increased, and, conversely, in case outstanding shares of Common Stock shall each be combined into a smaller number of shares of Common Stock, the number of shares of Common Stock issuable upon exercise of this Warrant at the opening of business on the day following the day upon which such combination becomes effective shall be proportionately decreased, such increase or decrease, as the case may be, to become effective immediately after the opening of business on the day following the date upon which such subdivision or combination becomes effective.

(3) Reclassifications. The reclassification of Common Stock into securities (other than Common Stock) and/or cash and/or other consideration shall be deemed to involve a subdivision or combination, as the case may be, of the number of shares of Common Stock outstanding immediately prior to such reclassification into the number or amount of securities and/or cash and/or other consideration outstanding immediately thereafter and the effective date of such reclassification shall be deemed to be “the day upon which such subdivision becomes effective,” or “the day upon which such combination becomes effective,” as the case may be, within the meaning of clause (2) above.

(4) Optional Adjustments. Columbia may make such increases in the number of shares of Common Stock issuable upon exercise of this Warrant, in addition to those required by this subparagraph (A), as shall be determined by its Board of Directors to be advisable in order to avoid taxation so far as practicable of any dividend of stock or stock rights or any event treated as such for federal income tax purposes to the recipients, subject to applicable Nasdaq stockholder approval requirements.

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(5) Adjustment to Exercise Price. Whenever the number of shares of Common Stock issuable upon exercise of this Warrant is adjusted as provided in this Paragraph 6(A), the Exercise Price shall be adjusted by a fraction in which the numerator is equal to the number of shares of Common Stock issuable prior to the adjustment and the denominator is equal to the number of shares of Common Stock issuable after the adjustment, rounded to the nearest cent.

(B) Certain Sales of Common Stock.

(1) Adjustment to Shares Issuable. If and whenever Columbia sells or otherwise issues (other than under circumstances in which Paragraph 6(A) applies) any shares of Common Stock, the number of shares of Common Stock issuable upon exercise of this Warrant shall be increased by multiplying such number of shares by a fraction, the denominator of which shall be the number shares of Common Stock outstanding at the close of business on the day immediately preceding the date of such sale or issuance and the numerator of which shall be the sum of such number of shares and the total number of shares constituting such sale or other issuance, such increase to become effective immediately after the opening of business on the day following such sale or issuance.

(2) Adjustment to Exercise Price. If and whenever Columbia sells or otherwise issues any shares of Common Stock (excluding any stock dividend or other issuance not for consideration to which Paragraph 6(A) applies or shares of Common Stock issued or issuable in connection with awards granted under the Columbia Stock Option Plans) for a consideration per share which is less than the Exercise Price at the time of such sale or other issuance, then in each such case the Exercise Price shall be forthwith changed (but only if a reduction would result) to the price (calculated to the nearest cent) determined by dividing: (i) an amount equal to the sum of (aa) the number of shares of Common Stock outstanding immediately prior to such issue or sale, multiplied by the then effective Exercise Price, plus (bb) the total consideration, if any, received and deemed received by Columbia upon such issue or sale, by (ii) the total number of shares of Common Stock outstanding immediately after such issue or sale.

(C) Definition. For purposes of this Paragraph 6, the term “Common Stock” shall include (1) any shares of Columbia of any class or series which has no preference or priority in the payment of dividends or in the distribution of assets upon any voluntary or involuntary liquidation, dissolution or winding up of Columbia and which is not subject to redemption by Columbia, and (2) any rights or options to subscribe for or to purchase shares of Common Stock or any stock or securities convertible into or exchangeable for shares of Common Stock (such convertible or exchangeable stock or securities being hereinafter called “Convertible Securities”), whether or not such rights or options or the right to convert or exchange any such Convertible Securities are immediately exercisable. For purposes of any adjustments made under Paragraph 6(A) or 6(B) as a result of the distribution, sale or other issuance of rights or options or Convertible Securities, the number of Shares of Common Stock outstanding after or as a result of the occurrence of events described in Paragraph 6(A)(1) or 6(B)(1) shall be calculated by assuming that all such rights, options or Convertible Securities have been exercised for the maximum number of shares issuable thereunder.

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7. Notice.

(A) Whenever the number of shares of Common Stock for which this Warrant is exercisable is adjusted as provided in Paragraph 6, Columbia shall promptly compute such adjustment and mail to the Holder a certificate, signed by the principal financial officer of Columbia, setting forth the number of shares of Common Stock for which this Warrant is exercisable as a result of such adjustment having become effective.

(B) Upon the occurrence of any event which results in the Holder having the right to require Columbia to repurchase this Warrant, as provided in Paragraph 7 of the Warrant Agreement, Columbia shall promptly notify the Holder of such event; and Columbia shall promptly compute the Warrant Repurchase Price, as adjusted by Paragraph 9 of the Warrant Agreement, and furnish to the Holder a certificate, signed by the principal financial officer of Columbia, setting forth the Warrant Repurchase Price, as adjusted, and the basis and computation thereof.

8. Rights of the Holder.

(A) Without limiting the foregoing or any remedies available to the Holder, it is specifically acknowledged that the Holder would not have an adequate remedy at law for any breach of the provisions of this Warrant and shall be entitled to specific performance of Columbia’s obligations under, and injunctive relief against any actual or threatened violation of the obligations of any Person (as defined in Paragraph 7 of the Warrant Agreement) subject to, this Warrant.

(B) The Holder shall not, by virtue of its status as Holder, be entitled to any rights of a stockholder in Columbia.

9. Termination. This Warrant and the rights conferred hereby shall terminate (i) upon the Effective Time of the Merger provided for in the Merger Agreement, (ii) upon a valid termination of the Merger Agreement (except a termination pursuant to Section 8.1(b)(iii) of the Merger Agreement) unless an event described in Paragraph 2 of the Warrant Agreement permitting exercise of the Warrant occurs prior to such termination, in which case this Warrant and the rights conferred hereby shall not terminate until 12 months after the occurrence of such event, (iii) to the extent this Warrant has not previously been exercised, 12 months after the occurrence of an event described in Paragraph 2 of the Warrant Agreement permitting exercise of the Warrant (unless termination of the Merger Agreement in accordance with its terms (other than under Section 8.1(b)(iii) thereof) occurs prior to the occurrence of such event, in which case (ii) above shall apply) or (iv) 12 months after the occurrence of a valid termination under Section 8.1(b)(iii) of the Merger Agreement.

10. Governing Law. This Warrant shall be deemed to have been delivered in, and shall be governed by and interpreted in accordance with the substantive laws of, the Commonwealth of Pennsylvania, except to the extent that Maryland law governs certain aspects of this Warrant as it relates to Columbia. In the event of any inconsistency between this Warrant and the terms of the Warrant Agreement, the terms of the Warrant Agreement shall govern.

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Dated: July 26, 2005

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